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Written by: Larry E. Collins
Geometry:A Complete Course
(with Trigonometry)
Module C – Solutions Manual
ERRATA
4/2010
Unit III — Fundamental TheoremsPart A — Deductive Proof
p. 215 – Lesson 1 — Direct Proof
1. a) Reflexive Property of Equality f) Multiplication Property of Equality
b) Transitive Property of Equality g) Distributive Property of Multiplication to Addition
c) Symmetric Property of Equality h) Addition Property of Equality
d) Addition Property for Equations i) Substitution Principle
e) Multiplication Property for Equations j) Property of Zero for Addition
2. b) Addition Property for Equations g) Multiplication Property for Equations
c) Arithmetic Fact h) Associative Property for Multiplication
d) Commutative Property for Addition i) Multiplicative Inverse Property
e) Additive Inverse Property j) Property of One for Multiplication
f) Property of Zero for Addition k) Substitution
3. b) Multiplication Property for Equations e) Property of One for Multiplication
c) Associative Property for Multiplication f) Substitution
d) Multiplicative Inverse Property
4. b) Multiplication Property for Equations f) Addition Property of Equality
c) Multiplicative Inverse Property g) Additive Inverse Property
d) Property of One for Multiplication h) Property of Zero for Addition
e) Substitution i) Substitution
5. 1. Given 4. Postulate 7 - Protractor - Fourth Assumption
2. Given 5. Postulate 7 - Protractor - Fourth Assumption
3. Addition Property for Equality 6. Substitution
6. 2. ST; Postulate 6 - Ruler - Fourth Assumption 6. Addition Property of Equality
3. RN; Postulate 6 - Ruler - Fourth Assumption 7. Additive Inverse Property
4. Substitution Principle 8. Property of Zero for Addition
5. Given
7. 2. Definition of Segment Congruence 6. Postulate 6 - Ruler - Fourth Assumption
3. Reflexive Property of Equality 7. FA = LT
4. Addition Property for Equality 8. Definition of Segment Congruence
5. Postulate 6 - Ruler - Fourth Assumption
8. Conditional: If 2x + 8 = – 16, then x = – 12.
Given: 2x + 8 = – 16 Prove: x = – 12
Statement Reason
1. 2x + 8 = – 16 1. Given
2. 2. Addition Property of Equality.
3. 3. Additive Inverse Property
4. 4. Property of Zero for Addition
5. 2x = – 24 5. Substitution
6. 6. Multiplication Property for Equations
7. 7. Associative Property of Multiplication
8. 8. Multiplicative Inverse Property
1Part A – Deductive Proof
2x + 8 + -8 = -16 + -8
2x + 0 = -16 + -8
2x = -16 + -8
12
12
2 = -24i i
( )
( )x
12
12
( )
( ) 2 = -24x
1 =
12
-24i x
( )
2 Unit III – Fundamental Theorems
8.Continued
Statement Reason
9. 9. Property of One for Multiplication
10. x = -12 10. Substitution
9. Conditional: In nRST, if /1 > /3 and /2 > /4, then /SRT > /STR
Given: /1 > /3 Prove: /SRT > /STR
/2 > /4
Statement Reason
1. /1 > /3 1. Given
/2 > /4
2. m/1 = m/3 2. Definition of Congruence
m/2 = m/4
3. m/1 + m/2 = m/3 + m/4 3. Addition Property for Equality
4. m/1 + m/2 = m/SRT 4. Postulate 7 - Protractor - Fourth Assumption
m/3 + m/4 = m/STR
5. m/SRT = m/STR 5. Substitution
6. /SRT > /STR 6. Definition of Congruence
10. Conditional: In nRST, if RQ > TP and MQ > MP, then RM > TM.
Given: RQ > TP Prove: RM > TM
MQ > MP
Statement Reason
1. RQ > TP , MQ > MP 1. Given
2. RQ = TP 2. Definition of Congruence
3. RM + MQ = RQ 3. Postulate 6 - Ruler (Segment–Addition Assumption)
TM + MP = TP
4. RM + MQ = TM + MP 4. Substitution (statements 3 and 1)
5. MQ = MP 5. Definition of Congruence
6. RM + MQ = TM + MP 6. Substitution
7. RM + MQ - MQ = TM + MP -MP 7. Subtraction Property of Equality
8. RM + 0 = TM + 0 8. Additive Inverse Property
9. RM = TM 9. Property of Zero for Addition
10. RM > TM 10. Definition of Congruence
x = 12
-24
( )
Unit III — Fundamental TheoremsPart A — Deductive Proof
p. 223 – Lesson 2 — Indirect Proof
1. a) The sun is not shining.
b) MN > PQ
c) /A is acute.
d) nABC is not congruent to nDEF.
2. a) not a contradiction
b) contradiction
c) not a contradiction
d) not a contradiction
3. a) Suppose the light bulb is not defective.
b) Suppose the number has more than two integer factors.
c) Suppose the two angles are not congruent.
d) Suppose A, B, and C are not collinear.
e) Suppose the two lines are not skew lines.
4. a) Conditional - If line , passes through point P and is parallel to line k, and line m passes through point P and
is parallel to line k, then line , and line m are the same line.
b) Diagram-
c) Given - , is a line that passes through point P.
m is a line that passes through point P.
, II k and m Il k.
d) Prove: , and mare the same line.
e) Proof:
1. Suppose line , and line m are not the same line. 1. Indirect Proof Assumption
2. Line , passes through point P and is parallel 2. Given
to line k.
3. Line m passes through point P and is parallel 3. Given
to line k.
4. There are two lines through point P parallel 4. Steps 2 and 3
to line k.
5. Step 4 is a contradiction. 5. Unit II - Postulate 9 - In a plane, through a point not on a given
line, there is exactly one line parallel to the given line.
6. Our assumption is false, line , and line m 6. R.A.A.
are the same line.
5. a) Conditional - If /AXY and /YXB form a linear pair and m/AXY > 90, then /YXB is not a right angle.
b) Diagram-
c) Given - /AXY and /YXB form a linear pair; m/AXY > 90
d) Prove - /YXB is not a right angle.
l
3Part A – Deductive Proof
m
k
P
A X B
Y
>90º
5. - continued
e) Proof
1. Suppose /YXB is a right angle. 1. Indirect Proof Assumption
2. m/YXB = 90O 2. A right angle is an angle whose measure is 90º
3. /AXY and /YXB form a linear pair. 3. Given
4. AX and XB are opposite rays. 4. Definition of Linear Pair
5. /AXB is a straight angle 5. Definition of Straight Angle
6. m/AXB is 180 6. Definition of Straight Angle
7. /AXY and /YXB are supplementary. 7. Defintion of supplementary angles.
8. m/AXY + m/YXB = 180O 8. Definition of supplementary angles
9. m/AXY + 90 = 180 9. Substitution
10. m/AXY + 90 + -90 = 180 + -90 10. Addition Property of Equality
11. m/AXY + 0 = 180 + -90 11. Additive Inverse Property Number Fact
12. m/AXY = 180 + -90 12. Identity Property for Addition
13. m/AXY = 90 13. Substitution
14. However, m/AXY > 90 14. Given
15. Steps 10 and 11 represent a contradiction. m/AXY 15. Postulate 7 - Protractor - Second Assumption - cannot both equal 90 and be greater than 90. an angle has a unique measure
16. Our original assumption must be false. 16. R.A.A.m/YXB > 90 and therefore, not a right angle.
6. a) Conditional: If a triangle is a right triangle, then it has no more than two acute angles.
b) Diagram
c) Given - nABC is a right triangle with right angle C.
d) Prove - nABC has no more than two acute angles
e) Proof
1. Suppose nABC has three acute angles, 1. Indirect Proof Assumption
/A, /B, and /C
2. m/C < 90O 2. Definition of Acute Angle
3. nABC is a right triangle with right angle C. 3. Given
4. m/C = 90O 4. A right angle is an angle whose measure is 90º
5. Steps 2 and 4 represent a contradiction. 5. Postulate 7 - Protractor - Second Assumption -
m/C cannot both equal 90 and be less than 90 measure of an angle is a unique real number.
6. Our original assumption must be false. 6. R.A.A.
nABC is a right triangle with no more than
two acute angles.
4 Unit III – Fundamental Theorems
A
C B
Unit III — Fundamental TheoremsPart B — Theorems about Points and Lines
p. 228 – Lesson 1 — Theorem 1: If a point lies outside a line, then exactly one plane contains the line and the point.
1. a) If a given point lies outside a given line, then exactly one plane contains the given line and the given point.
b)
c) Given: Point N lies outside line ,d) Prove: Exactly one plane contains point N and line ,.
e)
Statement Reason
1. Point N lies outside line , 1. Given
2. Points A and B are on line , 2. Every line contains at lest two points. (Postulate 1)
3. There is exactly one plane containing 3. For any three different non-collinear points,
points A,B, and N. there is EXACTLY ONE PLANE containing them.(Postulate 2)
4. line , is in plane P 4. For any two different points in a plane, the line
containing them is in the plane.
5. exactly one plane contains the given 5. Q.E.D.
line and the given point.
2.
3. Yes; We know from Theorem1, that point A and line DC are in exactly one plane. (If a point (point A) lies outside a line(Line DC or
Line DB on Line CB), then exactly one plane contains the line and the point.) Since the points D, C, and B are on Line DC, the
points are all in the same plane as the line. Points A, B, C, and D lie in the same plane.
4. Conditional: If nABC, with side BC in BD, has interior /ACB congruent to /ACD, then nABC is a right triangle.
a) I can prove triangle ABC is a right triangle if I can prove triangle ABC has a right angle. I can prove /ACB measures 90O. I can
prove this if I can prove m/ACB is half the measure of a straight angle which measures 180O. The rays, CB and CD are
opposite rays which makes /BCD a straight angle, so I can do this plan. Start at the end of the analysis and work backwards.
b) Statement Reason
1. nABC with side BC on BD 1. Given
2. /BCD is a straight angle. 2. Definition of a Straight Angle -
sides of angle are opposite rays by observation
3. m/BCD = 180 3. The measure of a straight angle is 180º.
4. m/BCA + m/ACD = m/BCD 4. Postulate 7 - Protractor - Fourth Assumption
5. m/ACB + m/ACD = 180 5. Substitution
6. /ACD > /ACB 6. Given
7. m/ACD = m/ACB 7. Definition of Congruent Angles
8. m/BCA + m/ACB = 180° 8. Substitution
9. (1 + 1) • m/ACB = 180° 9. Distributive Property of Multiplication to Addition
ll
The figure appears to be
a right triangle.
N
A
B
B C D
A
5Part B – Theorems About Points and Lines
4. b) continued.
Statement Reason
10. 2 • m/ACB = 180O 10. Substitution
11. 11. Multiplication Property of Equality
12. 12. Multiplicative Inverse Property
13. 1 • m/ACB = 90O 13. Substitution
14. m/ACB = 90 14. Multiplication Property of One
15. /ACB is a right angle 15. Definition of Right Angle
16. nACB is a right angle 16. Definition of Right Triangle
5. a) I can prove /ABD and /CBD are complimentary if I can prove the sum of their measures is 90O. I can prove their sum is 90 if I
can prove m/ABC is 90O. I know /3 is a right angle (90O) and forms a linear pair with /4 (which makes /4
equal to 90O). /4 is congruent to /ABC, so I can do this plan. Start at the end of the analysis and work backwards.
b) Statement Reason
1. /ADC is a straight angle. 1. Definition of Straight Angle- sides of angle are opposite rays
2. m/ADC = 180 2. The measure of a straight angle is 180º.
3. m/3 + m/4 = m/ADC 3. Postulate 7 - Protractor - Fourth Assumption
4. /3 is a right angle 4. Given
5. m/3 = 90 5. The measure of a right angle is 90º
6. 90 + m/4 = 180 6. Substitution of Equality (5 and 2 into 3)
7. 90 + m/4 + -90 = 180 + -90 7. Addition Property for Equations
8. 90 + -90 + m/4 = 180 + -90 8. Commutative Property of Addition
9. 0 + m/4 = 180 + -90 9. Additive Inverse Property
10. m/4 = 180 + -90 10. Property of Zero for Addition
11. m/4 = 90 11. Arithmetic Fact or Substitution
12. /4 > /ABC 12. Given
13. m/4 = m/ABC 13. Definition of Congruent Angles
14. 90 = m/ABC 14. Substitution
15. m/ABD + m/DBC = m/ABC 15. Postulate 7 - Protractor - Fourth Assumption
16. m/ABD + m/DBC = 90 16. Substitution
17. /ABD and /DBC are complementary angles 17. Definition of Complementary Angles
(sum of Two Angle measures is 90 degrees)
1
2 2 m ACB =
1
2 180O⋅ ⋅ ∠ ⋅
6 Unit III – Fundamental Theorems
1 m ACB =
1
2 180O⋅ ∠ ⋅
6. a) Statement Reason
1. AB > CD 1. Given
2. AB = CD 2. Definition of Congruent Segments
3. BC = BC 3. Reflexive Property of Equality
4. AB + BC = BC + CD 4. Additional Property for Equations
5. AB + BC = AC 5. Postulate 6 - Ruler - Fourth Assumption
BC + CD = BD
6. AC = BD 6. Substitution Principle (5 into 4)
7. AC > BD 7. Definition of Congruent Segments
b) Statement Reason
1. AC > BD 1. Given
2. AC = BD 2. Definition of Congruent Segment
3. AB + BC = AC 3. Postulate 6 - Ruler - Fourth Assumption
BC + CD = BD
4. AB + BC = BC + CD 4. Substitution
5. BC = BC 5. Reflexive Property of Equality
6. AB + BC - BC = BC + CD - BC 6. Subtraction Property of Equality
7. AB + BC - BC = CD + BC - BC 7. Commutative Property of Addition
8. AB + 0 = CD + 0 8. Additive Inverse Property - by definition,
subtraction means to “add the opposite”.
9. AB = CD 9. Property of Zero for Addition
10. AB > CD 10. Definition of Congruent Segments
7. a) AC > BD b) WX > YZ c) MO > NP
8. a) Statement Reason
1. AB > BD 1. Given
2. BC > DE 2. Given
3. BD > CE 3. Exercise 6 - Part A - Common Segment
4. AB = BD, BD = CE 4. Definition of Congruent Segments
5. AB = CE 5. Transitive Property of Equality
6. AB > CE 6. Definition of Congruent Segments
9. No, Theorem 1 states “If a point lies outside a line, then exactly one plane contains the line and the point.” Point A and collinear
points B, C, D, and E will be on exactly one plane by Theorem1. Point F and collinear points B, C ,D, and E will be on exactly one plane
by Theorem1. However, point A and point F are not necessarily on the same plane, so the planes ABCDE and FBCDE could form a
“vee” or dihedral angle with collinear points B, C, D, and E on the “edge” of the angle.
10. No, The proof only deals with the collinear points B, C, D, and E and the segment AB. The segments CF and FE have no
connection to the proof.
7Part B – Theorems About Points and Lines
Unit III — Fundamental TheoremsPart B — Theorems About Points and Lines
p. 231 – Lesson 2 — Theorem2: If three different points are on a line, then at most one is between the other two
1. a) If three different points are on a line, then at most one is between the other two.
b)
c) Given: points A, B, and C on line ,d) Prove: B alone lies between A and C
e) Statement Reason
1. Suppose point B is between points A and C, 1. Indirect Proof Assumption.
and point C is between points A and B.
2. AB + BC = AC 2. Postulate 6 - Ruler - Fourth Assumption
3. AC + CB = AB 3. Postulate 6 - Ruler - Fourth Assumption
4. (AC + BC) + BC = AC 4. Substitution (3 into 2)
5. AC + (BC + BC) = AC 5. Associative Property of Addition
6. AC + (1 + 1) BC = AC 6. Distributive Property of Multiplication to Addition -
7. AC + 2BC = AC 7. Substitution
8. AC + 2BC + -AC = AC + -AC 8. Addition Property for Equations
9. AC + -AC + 2BC = AC + -AC 9. Commutative Property of Addition
10. 0 + 2BC = 0 10. Additive Inverse Property
11. 2BC = 0 11. Identity Property of Addition
12. 12. Multiplication Property of Equality
13. 13. Multiplicative Inverse Property
14. 14. Multiplication Property of 1
15. BC = 0 15. Multiplication Property of Zero (or Substitution)
16. B and C must be the same point 16. Since BC = 0
17. Points A, B, and C are on line , 17. Given
18. This contradiction means our 18. B cannot be between A and C at the same time
assumption is false. C is between A and B.
19. B alone lies between A and C 19. R.A.A.
2. a) line b) line c) plane d) plane e) plane
3. a) always b) never c) always d) sometimes e) never
l
1
2
1
2 2 BC = 0i i i
1
1
20i iBC =
BC =
1
20i
A
B
C
8 Unit III – Fundamental Theorems
4. 5. 6.
7.
8. Statement Reason
1. EX > WG 1. Given
2. EX = WG 2. Definition of Congruent Segments
3. EW + WX = EX 3. Postulate 6 - Ruler - Fourth Assumption
WX + XG = WG
4. EW + WX = WX +XG 4. Substitution
5. EW + WX + -WX = WX + XG + -WX 5. Addition Property of Equality
6. EW + WX + -WX = WX + -WX + XG 6. Commutative Property of Addition
7. EW + 0 = 0 + XG 7. Additive Inverse Property
8. EW = XG 8. Identity Property of Addition
9. GX > HX 9. Given
10. GX = HX 10. Definition of Congruent Segments
11. EW = HX 11. Transitive Property of Equality
(step 8 and 10 - EW = XG, GX = HX, so EW = HX)
12. EW > FW 12. Given
13. EW = FW 13. Definition of Congruent Segments
14. FW = EW 14. Symmetric Property of Equality
15. FW = HX 15. Transitive Property of Equality (Step 14 to 11)
16. FW > HX 16. Definition of Congruent Segments
BD - BC = CD
12 - 4 = CD
8 = CCD
AB CD
AB CD
AB
≅=
8=
W + Y = WY
+ 2 + 1
X X
X 22 = 3 + 6
+ 14 = 3 + 6
X
X X
X + - + 14 + -6 = 3X + - + 6 + -6
X X
0 + 8 = 2 + 0
X
8 = 2
1
2
1
2i i X
4 =
WY Z
X
X≅ WY Z
+ 2 + 12 =
= X
X Z
+ 14 = Z
X
X X
18 = Z
Y + YZ
X
X == Z
12 + YZ = 18
12 + Y
X
ZZ + -12 = 18 + -12
0 + YZ == 6
YZ = 6
FG + GH = FH
24 + 6 = FH
30 = FH
EG FH
EG = F
≅HH
EG = 30
QR + RS = QS
8 + 2 = 4 - 2
8 + 2 + 2 + -2 = 4 - 2 +
x x
x x x 22 + -2
+ 0 = 2 + 0
x
x10
10 2
1
2
1
2i i= x
5 X
PQ + QR = PR
=
PQ RS
≅ PQ RS
PQ
== 2
PQ 2 5 = 10
x
= i 10 + 8 PR
= 18 PR=
9Part B – Theorems About Points and Lines
9. Statement Reason
1. GI > HJ 1. Given
2. GI = HJ 2. Definition of Congruent Segments
3. GH + HI = GI 3. Postulate 6 - Ruler - Fourth Assumption
HI + IJ = HJ
4. GH + HI = HI + IJ 4. Substitution Property
5. 5. Addition Property of Equality
6. 6. Commutative Property of Addition
7. GH + 0 = IJ + 0 7. Additive Inverse Property
8. GH = IJ 8. Identity Property of Equality
9. IK > JL 9. Given
10. IK = JL 10. Definition of Congruent Segments
11. IJ + JK = IK 11. Postulate 6 - Ruler - Fourth Assumption
JK + KL = JL
12. IJ + JK = JK + KL 12. Substitution
13. IJ + JK + -JK = JK + KL + JK 13. Addition Property of Equality
14. IJ + JK + -JK = JK + -JK + KL 14. Commutative Property of Addition
15. IJ + 0 = 0 + KL 15. Additive Inverse Property
16. IJ = KL 16. Identity Property of Addition
17. GH = KL 17. Transitive Property of Equality (Statement 8 - Statement 16)
18. GH > KL 18. Definition of Congruent Segments
10. Statement Reason
1. /LMQ > /NMP 1. Given
2. m/LMQ = m/NMP 2. Definition of Congruent Angles
3. m/LMP + m/PMQ = m/LMQ 3. Postulate 7 - Protractor - Fourth Assumption
4. m/NMQ + m/QMP = m/NMP 4. Postulate 7 - Protractor - Fourth Assumption
5. m/LMP + m/PMQ = m/NMQ + m/QMP 5. Substitution
6. m/LMP + m/PMQ + -m/PMQ = 6. Addition Property of Equalitym/NMQ + m/QMP + -m/QMP
7. m/LMP + 0 = m/NMQ + 0 7. Additive Inverse Property
8. m/LMP = m/NMQ 8. Identity Property of Addition
9. /LMP > /NMQ 9. Definition of Angle Congruence.
GH + HI + -HI HI + IJ + -HI=
GH + HI + -HI IJ + HI + -HI=
10 Unit III – Fundamental Theorems
Unit III — Fundamental TheoremsPart C — Theorems About Segments and Rays
p. 234 – Lesson 1 — Theorem 3: If you have a given ray, then there is exactly one point, at a given distance from the endpoint of the ray.
1. a) If you have a given ray, then there is exactly one point, at a given distance from the end point of the ray.
b)
c) Given: AB and distance d.
d) Prove: Only X is at a distance d from point A on AB
e) Statement Reason
1. AB 1. Given.
2. Zero corresponds to point A. 2. Postulate 6 - Ruler Postulate - First Assumption
3. All other points on the ray correspond 3. Postulate 6 - Ruler Postulate - First Assumption
to positive real numbers. X and B correspond
to positive real numbers.
4. We have “d ” 4. Given
5. | O - X | = d 5. Postulate 6 - Ruler Postulate - Third Assumption
6. Only one point corresponds to d 6. Postulate 6 - Ruler Postulate - Second Assumption
(distance is unique)
2. One
3. Point Q
4. No: The rays do not have the same initial (or end) point, and could go in opposite directions.
5. No.
Case 1: If point T is between point Q and point S, or point S is between point Q and point T, then the rays are the same.
Case 2: Q, T, and A, may not be collinear.
6. No; If point T is between point Q and Point S, or point S is between point Q and point T, then the rays are the same.
11Part C – Theorems About Segments and Rays
A X B
O X b
Q T S Q S T
Q T S Q S T
T
S
Q
7. a)
b)
c)
d)
8. a) yes b) no
9.
BD AB + BD = AD DC AB + BC = z + x = ACBD = AD - AB AB + BD = yBD = y - z AC - (AB + BD) = DC
(z + x) -y = DCz + x -y = DC
10. a)
b)
c) – 9 — (– 3) = – 9 + 3 = – 6 = 6
11. 1. Given
3. Postulate 6 - Ruler - Fourth Assumption
5. Substitution (4 into 3)
8. Addition Property for Inequalities
9. Additive Inverse
10. Identity for Addition
11. Given
15. Addition for Inequality
16. Additive Inverse
17. Identity for Addition
21. Distributive Inequality
23. Multiplication Property for Inequalities
24. Property of Zero for Multiplication
25. Multiplication Inverse Property
26. Identity Multiplication
27. a < b means b > a
12 Unit III – Fundamental Theorems
Q A B
X A B
R S T
A B D
A B D
A B D
– 12 – 6 0
– 12 – 6
– 12 or 0
– 3 or – 90
C
y
z x
P Q
T
midpointmidpoint
28. Multiplication for Inequality
29. Distributive • to +
30. Multiplication for -1
31. Addition for Inequality
32. Additive Inverse
33. Identity Addition
35. Substitution
36. Additive Inverse
37. Multiplication for Inequality
38. Multiplication Property of Zero
39. Multiplication for -1
12. Statement Reason
1. X is between R and T; the coordinates of R 1. Given
and T are zero and t, respectively.
2. X has a coordinate, call it x 2. Postulate 6 - Ruler - First Assumption
3. RX + XT = RT 3. Postulate 6 - Ruler - Fourth Assumption
4. RX = 0 — x = — x = x 4. Postulate 6 - Ruler - Third Assumption
XT = x — t
RT = 0 — t = – t = t
5. x + x — t = t 5. Substitution (4 into 3)
6. x — t < t 6. Definition of >
7. x — t < t and x — t > — t 7. Algebra - If x < k, then x < k and x > – k.
8. t = t 8. Definition of Absolute Value
9. x — t < t and x — t > – t 9. Substitution
Consider x — t > — t
10. 10. Multiplication for Inequality
11. 11. Definition of Subtraction
12. 12. Distributive x to +
13. 13. Multiplication for – 1
14. 14. Addition for Inequality
15. -x + 0 < 0 15. Additive Inverse Property
16. -x < 0 16. Identity for Addition
17. 17. Multiplication for Inequality
18. 18. Property of -1 for Multiplication
19. x > 0 19. Property of Zero for Multiplication
20. The coordinate of X is positive. 20. Definition of Positive (Greater than Zero)
13Part C – Theorems About Segments and Rays
- - < - (-1 1( ) )x t t+ i
- < - (-1 1( ) )x t t− i
- (- - < - (-1 1 1i ix t t+ ) ( ) )
- < x t t+
- - < - x t t t t+ + +
(- (- (- 0 1 1) ) )x > i
x (- 0 > 1) i
12. (continued)
Consider x - t < t
21. x — t + t < t + t 21. Addition for Inequality
22. x + 0 < t + t 22. Additive Inverse Property
23. x < t + t 23. Identity for Addition
24. x < (1 + 1) t 24. Distributive x to 0
25. x < 2t 25. Substitution
26. t > 0 26. Given
27. t is a positive number 27. A number greater than zero is positive
28. x < a positive number 28. Substitution (x < 2 • positive number)
29. x could be zero or x could be negative 29. Zero or a negative number is less than a positive number.
30. x cannot be zero or negative 30. It is given that x is between zero and t. So, we have a
contradiction and cannot use this part of the “and” statement.
Unit III — Fundamental TheoremsPart C — Theorems About Segments and Rays
p. 239 – Lesson 2 — Theorem 4: If you have a given line segment, then that segment has exactly one midpoint.
1. a) Theorem 4: If you have a given line segment, then that segment has exactly one midpoint.
b)
c) Given: AB with midpoint C
d) Prove: C is the only midpoint of AB
e) Statement Reason
1. Line segment AB point C is the midpoint of AB 1. Given.
2. AC > CB 2. Definition of Midpoint
3. AC = CB 3. Definition of congruent segments
4. AC + CB = AB 4. Postulate 6 - Ruler Fourth Assumption
5. AC + AC = AB 5. Substitution (3 into 4)
6. 2 AC = AB 6. Collect like terms
7. 7. Multiplication for Equality
8. 8. Multiplicative Inverse Property
9. 9. Identity for Multiplication
10. Suppose D is a different midpoint of AB 10. Indirect Proof Assumption
11. AD > DB 11. Definition of Midpoint
12. AD = DB 12. Definition of Congruent Segments
A C B
1
2
1
2
=
2 AC ABi i
1
1
2 AC ABi i=
AC AB=
1
2i
14 Unit III – Fundamental Theorems
15Part C – Theorems About Segments and Rays
1. (continued)
13. AD + DB = AB 13. Postulate 6 - Ruler Fourth Assumption
14. AD + AD = AB 14. Substitution
15. 15. Distributive Property
16. 16. Collect Like Terms
17. 17. Multiplication for Equality
18. 18. Multiplicative Inverse
19. 19. Multiplicative Identity
20. AD = AC 20. Substitution (9 into 19)
21. C and D are the same point 21. Postulate 6 - Ruler - First Assumption
22. The segment has exactly one midpoint. 22. R.A.A.
2. Point Q must be between point M and point N so that m< q < n or n < q < m and MQ > QN (II-B-5)
3. AB = CD
4. Point N
5. Z is the midpoint of XY.
6. Midpoint of AM
7. CD ? EF
8. yes
9. Statement Reason
1. AB 1. Given
2. AB = AB 2. Reflexive Property of Equality
3. AB > AB 3. Definition of Congruent Line Segments
Reflexive property of congruence of line segments
10. Statement Reason
1. AB > CD 1. Given
2. AB = CD 2. Definition of Congruent Segments
3. CD = AB 3. Symmetric for Equality
4. CD > AB 4. Definition of Congruent Line Segments
Symmetric Property of Congruent Line Segments
(1 1) AD = AB+ i
2 AD = ABi
1
22
1
2 AD = ABi i
1
1
2 AD = ABi i
AD = AB
1
2i
11. Statement Reason
1. AB > CD 1. Given
CD > EF
2. AB = CD 2. Definition of Congruent Line Segments
CD = EF
3. AB = EF 3. Transitivity for Equality
4. AB > EF 4. Definition of Congruent Line Segments
Transitive Property of Congruent Line Segments
12. a)
b)
c) d)
13. 1. Given
2. Multiplication for Inequality
3. Property of Zero for Multiplication
4. Substitution
5. Definition of Distance or Postulate 6
6. Transitivity for Equality
7. Definition of Congruent Line Segments
8. Addition for Equality (5)
9. Distributive x to +
10. Substitution or Addition Fact
11. Identity for Multiplication
12. Substitution
13. Postulate 6 - Ruler - Fourth Assumption
14. Definition of Midpoint (7 and 13)
C M D
A
C
D
A
M
B
(answer can vary)
(answer can vary)
2 4 6C M D
12 15 18
l
16 Unit III – Fundamental Theorems
17Part C – Theorems About Segments and Rays
14. 1. Given
2. Postulate 6 - Ruler - First Assumption
3. Given
4. Subtraction for Inequality
5. Substitution
6. Definition of Midpoint
7. Definition of Congruent Line Segments
8. Definition of Distance or Postulate 6
9. Substitution (8 into 7)
10. Algebra: If x = k, then x = k or x = – k.
11. Addition for Equality
12. Distributivity x to +
13. Substitution of Equals
14. Commutativity for Addition
15. Additive Inverse
16. Identity for Addition
17. Multiplication for Equality
18. Multiplicative Inverse
19. Identity Property of One
20. Substitution
21. Substitution
Unit III — Fundamental TheoremsPart D — Theorems About Two Lines
p. 244 – Lesson 1 — Theorem5: If two lines intersect, then exactly one plane contains both lines.
1. a) If two lines intersect, then exactly one plane contains both lines.
b)
c) Given: , and m intersect at T
d) Prove: Exactly one plane contains , and m.
e) Statement Reason
1. , and m intersect at T 1. Given.
2. Point T lies on , and m 2. Definition of Intersecting Lines.
3. Point X lies on ,. 3. Every line contains at least two points.
Point Y lies on m.
4. Points X, Y, and T do not lie on the same line. 4. Step 4 Statement
5. Exactly one plane contains T, X and Y. 5. Exactly one plane contains three different non-collinear points.
6. , is in plane P. 6. Postulate 3 - for any two different points in a plane,
m is in plane P. the line contain them is in the plane.
7. Exactly one plane contains two intersecting lines. 7. Q.E.D.
2. a) False k) True
b) True l) True
c) True m) False
d) True n) True
e) False o) False
f) False p) True
g) True q) False
h) True r) True
i) False s) False
j) False t) True
3. Given: Line , and m intersect at point X.
/UXV > /VXW
Prove: Line , line m.
d) Statement Reason
1. , and m intersect at X 1. Given.
2. /UXW is a straight angle equal to 180º. 2. Definition of straight angle - an angle whose sides are
opposite rays, giving a measure of 180º.
18 Unit III – Fundamental Theorems
T
m
l
⊥
19Part D – Theorems About Two Lines
3. (continued)
Statement Reason
3. m/UXV + m/VXW = /UXW 3. Postulate 7 - Protractor - Fourth Assumption.
4. m/UXV + m/VXW = 180º 4. Substitution of Equals (2 into 3)
5. /UXV > /VXW 5. Given
6. m/UXV = m/VXW 6. Definition of Congruent angles.
7. m/UXV + m/UXV = 180 7. Substitution of Equals (6 into 3)
8. (1 + 1) • m/UXV = 180 8. Distributive Property of Multiplication Over Addition.
9. 2m/UXV = 180 9. Substitution of Equals (2 = 1 + 1)
10. 10. Multiplication Property for Equations
11. 11. Substitution of Equals
12. 1 • m/UXV = 90 12. Multiplicative Inverse Property
13. m/UXV = 90 13. Multiplicative Identity
14. /UXV is a right angle 14. Definition of right angle
15. , m 15. Definition of perpendicular lines - lines which intersect
to form a right angle
4. Given: Points R, S, T, and U are collinear
RT > SU
Prove: RS > TU
Statement Reason
1. Points R, S, T, and U are collinear 1. Given
2. RT > SU 2. Given
3. RS + ST = RT 3. Postulate 6 - Ruler - Fourth Assumption.
ST + TU = SU
4. RS + ST > ST + TU 4. Substitution of Equals (3 into 2)
5. RS + ST + -ST 5. Property of Addition for Inequalities
> ST + TU + -ST
6. RS + ST + -ST 6. Commutative Property of Addition
ST + -ST + TU
7. RS + 0 > 0 + TU 7. Property of Additive Inverse (a + -a = 0)
8. RS > TU 8. Identity Property of Addition (a + o = a)
5. Given: /YXZ and /YZX are complementary
/XYW > /YZX
/WYZ > /YXZ
Prove: XY YZ
Statement Reason
1. /YXZ and /YZX are complementary 1. Given
1
2180
180
2 = = 90i
1
22
1
2 UXV = 180i im∠
1
22 9 UXV = 0i m∠
5. (continued)
Statement Reason
2. m/YXZ + m/YZX = 90 2. Definition of Complementary Angles
3. /WYZ > /YXZ 3. Given
/XYW > /YZX
4. m/WYZ = m/YXZ 4. Definition of Congruent Angles
m/XYW = m/YZX
5. m/WYZ + m/XYW = 90 5. Substitution of Equals (4 into 2)
6. m/XYW + m/WYZ = 90 6. Commutative Property of Addition
7. m/XYW + m/WYZ = m/XYZ 7. Postulate 7 - Protractor - Fourth Assumption
8. m/XYZ = 908. Substitution of Equals (7 into 6)
9. /XYZ is a right angle 9. Definition of Right Angle
10. XY YZ 10. Definition of Perpendicular Lines -
Lines which intersect to form a right angle.
6. Given: m/EBC = m/ECB
Prove: m/ABE = m/ECD
Statement Reason
1. /ABC is a straight angle with a measure of 180º 1. Definition of Straight Angle - an angle whose sides are
/DCB is a straight angle with a measure of 180º opposite rays, giving a measure of 180º
2. m/ABE + m/EBC = m/ABC 2. Postulate 7 - Protractor - Fourth Assumption
m/BCE + m/ECD = m/DCB
3. m/ABE + m/EBC = 180 3. Substitution of Equals (1 into 2)
m/BCE + m/ECD = 180
4. m/ABE + m/EBC = 4. Substitution of Equals (3 into 3)
m/BCE + m/ECD
5. m/EBC = m/ECB 5. Given
6. m/ABE + m/EBC + -m/EBC = 6. Property of Addition for Equality
m/BCE + m/ECD + -m/ECB
7. m/ABE + m/EBC + -m/EBC = 7. Commutative Property of Addition
m/ECD + m/BCE + -m/ECB
8. m/ABE + 0 = m/ECD + 0 8. Property of Additive Inverse (a + -a = o)
9. m/ABE = m/ECD 9. Identity Property of Addition
7. Given: m/DFG = m/HBC = 180
Prove: m/HFG m/HBC
Statement Reason
1. /DFH is a straight angle with a measure of 180º 1. Definition of Straight Angle - an angle whose sides are
opposite rays, giving a measure of 180º
2. m/DFG + m/HFG = m/DFH 2. Postulate 7 - Protractor - Fourth Assumption
3. m/DFG + m/HFG = 180 3. Substitution of Equals (1 into 2)
4. m/DFG + m/HBC = 180 4. Given
20 Unit III – Fundamental Theorems
AB»
21Part D – Theorems About Two Lines
7. (continued)Statement Reason
5. m/DFG + m/HFG = m/DFG + m/HBC 5. Substitution of Equals (4 into 3)
6. m/DFG + m/HFG + -m/DFG = 6. Property of Addition for Equality
m/DFG + m/HBC + -m/DFG
7. m/HBC + m/DFG + -m/DFG = 7. Commutative Property of Addition
m/HBC + m/DFG + -m/DFG
8. m/HFG + 0 = m/HBC + 0 8. Property of Additive Inverse
9. m/HFG = m/HBC 9. Identity Property for Addition (a + o = a)
10. /HFG > /HBC 10. Definition of Angle Congruence
8. Given: AX and BY intersect at point Z as shown.
AX > BY
ZX > ZY
Prove: AZ > BZ
Statement Reason
1. AX and BY intersect at point Z as shown 1. Given.AX > BY
2. AX = BY 2. Definition of Congruent Segments
3. AZ + ZX = AX 3. Postulate 6 - Ruler - Fourth Assumption. BZ + ZY = BY
4. AZ + ZX = BZ +ZY 4. Substitution of Equals (3 into 2)
5. ZX > ZY 5. Given
6. ZX = ZY 6. Definition of Congruent Segments
7. -ZX = -ZY 7. Additive Inverse Property - for every real number a, there exists a unique real number -a such that...
8. AZ + ZX + -ZX = BZ +ZY + -ZY 8. Property of Addition for Equality (4 + 7)
9. AZ + 0 = BZ + 0 9. Additive Inverse Property (a + -a = 0)
10. AZ = BZ 10. Identity Property of Addition
11. AZ > BZ 11. Definition of Congruent Segments
9. Given: PQ and RS intersect at point T.
PQ RS
Prove: /PTS > /STQ
/PTS > /PTR
/RTQ > /QTS
/RTQ > /RTP
Statement Reason
1. PQ and RS intersect at point T. 1. Given.
PQ RS
2. /PTS is a right angle. 2. Definition of Perpendicular lines - lines
/STQ is a right angle. which intersect to form a right angle.
/RTQ is a right angle.
/RTP is a right angle.
9. (continued)
Statement Reason
3. m/PTS = 90 3. Definition of a Right Angle
m/STQ = 90
m/RTQ = 90
m/RTP = 90
4. m/PTS = m/STQ 4. Substitution of Equals (3 into 3)
m/PTS = m/PTR
m/RTQ = m/QTS
m/RTQ = m/RTP
5. /PTS > /STQ 5. Definition of Congruent Angles
/PTS > /PTR
/RTQ > /QTS
/RTQ > /RTP
10. Given: m/ANC = m/ACF
Prove: m/NCD = m/FCD
Statement Reason
1. m/ANC = m/ACF 1. Given.
2. m/ACF + m/FCD = m/ACD 2. Postulate 7 - Protractor - Fourth Assumption -
m/ACN + m/NCD = m/ACD “Angle Addition” Assumption
3. m/ACF + m/FCD = m/ACN + m/NCD 3. Substitution of Equals (3 into 3)
4. m/ACF = m/ACN 4. Symmetric Property for Equality
5. -m/ACF = -m/ACN 5. Additive Inverse Property - for every real number a, there
exists a unique real number -a such that...
6. m/ACF + m/FCD + -m/ACF = 6. Property of Addition for Equality (3 + 5)
m/ACN + m/NCD + -m/ACN
7. m/FCD + m/ACF + -m/ACF = 7. Commutative Property of Addition
m/NCD + m/ACN + -m/ACN
8. m/FCD + 0 = m/NCD + 0 8. Additive Inverse Property (a + -a = o)
9. m/FCD = m/NCD 9. Identity Property of Addition (a + o = a)
10. m/NCD = m/FCD 10. Symmetric Property for Equality
Unit III — Fundamental TheoremsPart D — Theorems About Two Lines
p. 247 – Lesson 2 — Theorem6: If, in a plane, there is a point on a line, then there is exactly one perpendicular to the line,through that point.
1. a) Theorem6: If, in a plane, there is a point on a line, then there is exactly one perpendicular to the line, through that point.
b) M
P
B
A
l
22 Unit III – Fundamental Theorems
23Part D – Theorems About Two Lines
1.(continued)
c) Given: Point P is on AB in Plane M
d) Prove: Exactly one line in plane M is perpendicular to AB through point P.
e) Statement Reason
1. Point P is on AB 1. Given.
2. There exists a line in plane M, through point P, 2. Postulate 7 - Protractor - Second Assumption
which forms a 90º angle with AB
3. Line , is perpendicular to AB through point P. 3. Definition of Perpendicular
4. Suppose a second line, ,, also forms a 90º 4. Indirect Proof Assumption
angle through point P.
5. Line , is perpendicular to AB through point P. 5. Definition of Perpendicular
6. Line , must be the same as line , 6. Postulate 7 - Protractor - First Assumption - one to one
correspondence with real numbers.
7. Exactly one line in plane M is perpendicular to 7. R.A.A.
AB through point P
2. a) MC MB or MC MA or MC AB or MC BA
b) MC ; AB
c) Line t and CM are the same line. Theorem6 states that for a point on a line there can be only one perpendicular to the line at
that point. Since the two lines are in the same plane, they would have to be the same line.
d) MR and CM are the same line. (Theorem6) All points of MR correspond to all points of CM. Point R is on CM.
e) ; Theorem5 states that if two different lines intersect, then exactly one plane contains both lines. Line p contains point M,
the intersection of AB and CM. Line p must then be perpendicular to both lines (Theorem6)
3. a) False; choose point B not on CD
b) True
c) False; Consider the segment in space. The segment could have an infinite number of perpendicular bisectors.
d) False;
There are an infinite number of lines in plane K perpendicular to AB at midpoint M of AB.
l l
l
l
A
M
K
B
1
2
3
4
4. Given: BA AD
BC CD
/2 > /4
Prove: /1 > /3
Statement Reason
1. BA AD 1. Given
BC CD
2. /BAD is a right angle 2. Definition of Perpendicular Lines (line segments)
/BCD is a right angle
3. m/BAD = 90º 3. Definition of Right Angle
m/BCD = 90º
4. m/1 + m/2 = m/BAD 4. Postulate 7 - Protractor - Fourth Assumption.
m/3 + m/4 = m/BCD
5. m/1 + m/2 = 90 5. Substitution of Equals (3 into 4)
m/3 + m/4 = 90
6. m/1 + m/2 = m/3 + m/4 6. Substitution of Equals (4 into 4)
7. /2 > /4 7. Given
8. m/2 = m/4 8. Definition of Angle Congruence
9. -m/2 = -m/4 9. Property of Additive Inverse - For every real number a,
there exists a unique real number -a, such that...
10. m/1 + m/2 + -m/2 = 10. Addition Property for Equality (6 + 9)
m/3 + m/4 + -m/4
11. m/1 + 0 = m/3 + 0 11. Additive Inverse Property (a + -a = o)
12. m/1 = m/3 12. Identity Property for Addition (a + o = a)
13. /1 > /3 13. Definition of Angle Congruence
5. Given: /ABD and /DBC are complementary;
/ADB > /ABC
Prove: BD AC
Statement Reason
1. /ABD and /DBC are complementary 1. Given
2. m/ABD + m/DBC = 90 2. Definition of Complementary Angles
3. m/ABD + m/DBC = m/ABC 3. Postulate 7 - Protractor - Fourth Assumption.
4. m/ABC = 904. Substitution of Equals (3 into 2)
5. /ADB > /ABC 5. Given
6. m/ADB = m/ABC 6. Definition of Congruent Angles
7. m/ADB = 907. Substitution of Equals
8. /ADB is a right angle 8. Definition of Right Angle
9. BD AC 9. Definition of Perpendicular Lines (line segments)
24 Unit III – Fundamental Theorems
25Part D – Theorems About Two Lines
6. Given: /CBG > /EFA
/CBG and /EFA are supplementary
Prove: CD AG
Statement Reason
1. /CBG and /EFA are supplementary 1. Given
2. m/CBG + m/EFA = 180 2. Definition of Supplementary Angles
3. /CBG > /EFA 3. Given
4. m/CBG = m/EFA 4. Definition of Congruent Angles
5. m/CBG + m/CBG = 180 5. Substitution of Equals
6. (1 + 1) • m/CBG = 180 6. Distributive Property of Multiplication Over Addition
7. 2m/CBG = 180 7. Substitution of Equals (6 into 7; 1 + 1 =2)
8. 8. Property of Multiplication for Equality
9. 9. Property of Multiplicative Inverse
10. 10. Identity Property for Multiplication - (1 • a = a)
11. m/CBG = 90 11. Substitution of Equals
12. /CBG is a right angle 12. Definition of Right Angle
13. CD AG 13. Definition of Perpendicular Lines
7. a) m/MNP = m/ONP since both are right angles formed by two perpendicular lines.
b) MO NP since /MNP and /ONP are two equal angles forming a straight angle. Each will equal 90º. Each will be a right angle.
8. 135;
The sum of the two obtuse angles must equal 270º if the non common sides form a right angle. (360 - 90 = 270). Two angles whose
sum is 270, cut in half by angle bisectors, would be 135.
9. yes;
0 into 11 ; 90 = 180
1
2i
1
22
1
2 CBG = 180i im∠
1
1
2 CBG = 180i im∠
m∠CBG = 180
1
2i
C
OA
B
non-common sides
10. a) yes
b) no
Unit III — Fundamental TheoremsPart E — Theorems About Angles – Part 1 (One Angle)
p. 250 – Lesson 1 — Theorem7: If, in a half-plane, there is a ray in the edge of the half-plane, then there is exactly one other raythrough the endpoint of the given ray, such that the angle formed by the two rays has a givenmeasure.
1. a) Theorem7: If, in a half-plane, there is a ray in the edge of the half-plane, then there is exactly one other ray through the
endpoint of the given ray, such that the angle formed by the two rays has a given measure.
b)
c) Given: AB in the edge of half-plane H.
d) Prove: There is only one ray, which forms an angle with AB such that the measure of the angle is r.
P
M
N
45
C
AB
H
r
B
A
A
C
0
30
20
40
50
6070
80 90 100110
120
130
140
150
160
17010
180A
O
10
180
170
160
150
140
130
120110
100 8070
60
50
40
30
20
0
B
C
D E
F
G
H
J
H
P
Y
x
z
l
26 Unit III – Fundamental Theorems
1. (continued)
e) Statement Reason
1. AB is a ray in the edge of a half-plane. 1. Given
2. The real number r is the desired angle measure. 2. Given
3. Let AB correspond to the number zero. 3. Postulate 7 – Protractor – First Assumption
4. There is only one ray, call it AC, which corresponds 4. Postulate 7 – Protractor – First Assumptionto the real number r .
5. m/BAC = r - 0 or 0 - r 5. Postulate 7 – Protractor – Third Assumption
6. r - 0 = r = r 6. Definition of Absolute Value
7. There is only one ray, such that the angle formed 7. Q. E. D.with the given ray, has a given measure.
2. Given: /ABC
Prove: /ABC > /ABC
Statement Reason
1. /ABC 1. Given
2. m/ABC = m/ABC 2. Reflexive Property of Equality
3. /ABC > /ABC 3. Definition of Congruent Angles
The reflexive property for congruence of angles describes this relationship.
3. Given: /ABC > /DEF
Prove: /DEF > /ABC
a) If /ABC > /DEF, then /DEF > /ABC
b) Statement Reason
1. /ABC > /DEF 1. Given
2. m/ABC = m/DEF 2. Definition of Congruent Angles
3. m/DEF = m/ABC 3. Symmetric Property of Equality
4. /DEF > /ABC 4. Definition of Congruent Angles
The symmetric property for congruence of angles describes this relationship.
4. Given: /ABC > /DEF/DEF > /GHI
Prove: /ABC > /GHI
a) If /ABC > /DEF and /DEF > /GHI, then /ABC > /GHI
b) Statement Reason
1. /ABC > /DEF 1. Given/DEF > /GHI
2. m/ABC = m/DEF 2. Definition of Congruent Anglesm/DEF = m/GHI
3. m/ABC = m/GHI 3. Transitive Property of Equality
4. /ABC > /GHI 4. Definition of Congruent Angles
The transitive property for congruence of angles describes this relationship.
27Part E – Theorems About Angles – Part 1 (One Angle)
5. m/APC 6. m/DPA 7. m/BPC 8. m/CPD 9. 87 + 29 = 116 10. 78 – 49 = 29
11.
c) m/ABC = 135°
d) Supplementary
e) Opposite
f) 180°
12. The sum must be 180°
13. Given: BC CD; /FBR > /DCF
/CBF > /FCB
Prove: m/CBR = 90
Statement Reason
1. BC CD 1. Given
2. /DCB is a right angle 2. Definition of Perpendicular Lines (Segment to Ray)
3. m/DCB = 90° 3. Definition of Right Angle
4. m/DCF + m/FCB = m/DCB 4. Postulate 7 – Protractor – Fourth Assumption
5. m/DCF + m/FCB = 90 5. Substitution (3 into 4)
6. /FBR > /DCF 6. Given/CBF > /FCB
7. m/FBR = m/DCF 7. Definition of Congruent Anglesm/CBF = m/FCB
8. m/FBR + m/CBF = 90 8. Substitution (7 into 5)
9. m/CBF + m/FBR = 90 9. Commutative Property of Addition
10. m/CBF + m/FBR = m/CBR 10. Postulate 7– Protractor – Fourth Assumption
11. 90 = m/CBR 11. Substitution (9 into 10)
12. m/CBR = 90 12. Symmetric Property of Equality
14. a) 140; Exercise 12 – non-common sides opposite rays with /DPB.
b) 40; Exercise 12 – non-common sides opposite rays with /APD.
c) Yes; Definition of congruent angles
d) No
e) Yes
28 Unit III – Fundamental Theorems
P
M
N
45
C
AB
H
r
B
A
A
C
0
30
20
40
50
6070
80 90 100110
120
130
140
150
160
17010
180A
O
10
180
170
160
150
140
130
120110
100 8070
60
50
40
30
20
0
B
C
D E
F
G
H
J
Unit III — Fundamental TheoremsPart E — Theorems About Angles – Part 1 (One Angle)
p. 253 – Lesson 2 — Theorem8: If in a half-plane, you have an angle, then that angle has exactly one bisector.
1. a) Theorem 8: If in a half-plane, you have an angle, then that angle has exactly one bisector.
b)
c) Given: OB is a bisector of /COA in half-plane J
d) Prove: OB is the only bisector of /AOC
e) Statement Reason
1. OB is a bisector of /AOC 1. Given
2. /AOB > /BOC 2. Definition of Angle Bisector
3. m/AOB = m/BOC 3. Definition of Congruent Angles
4. OB lies between OA and OC 4. Definition of Angle Bisector
5. m/AOB + m/BOC = m/AOC 5. Postulate 7 – Protractor – Assumption Four –Angle Addition Assumption
6. m/AOB + m/AOB = m/AOC 6. Substitution (3 into 5)
7. (1 + 1) • m/AOB = m/AOC 7. Distributive Property of Multiplication Over Addition
8. 2 • m/AOB = m/AOB 8. Substitution
9. 9. Multiplication Property for Equality
10. 10. Multiplicative Inverse Property
11. 11. Identity Property of Multiplication (1 • a = a)
12. Suppose we have another bisector of /AOC, 12. Indirect Proof Assumptioncall it OD
13. /AOD > /DOC 13. Definition of Angle Bisector
14. m/AOD = m/DOC 14. Definition of Congruent Angles
15. OD lies between OA and OC 15. Definition of Angle Bisector
16. m/AOD + m/DOC = m/AOC 16. Postulate 7 – Protractor – Assumption Four –Angle Addition Assumption
17. m/AOD + m/AOD = m/AOC 17. Substitution (14 into 16)
18. (1 + 1) m/AOD = m/AOC 18. Distributive Property of Multiplication Over Addition
19. 2 • m/AOD = m/AOC 19. Substitution (1 + 1 = 2)
20. 20. Multiplication Property for Equality
21. 21. Multiplicative Inverse Property
22. 22. Identity Property of Multiplication
23. m/AOB = m/AOD 23. Substitution (11 into 22)
24. OB = OD (same ray) 24. Postulate 7 – Protractor – Second Assumption – Unique Measure
25. OB is the only bisector of m/AOC 25. R. A. A.
29Part E – Theorems About Angles – Part 1 (One Angle)0
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2 m AOD = m AOCi i∠ ∠
m AOD = m AOC∠ ∠1
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1
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2 m AOB = m AOCi i∠ ∠
m AOB = m AOC∠ ∠
1
2i
2. /QSR; Definition of Angle Bisector
3. a) QSP b) Bisector
4. a) 60° b) 75° c) 70°
5. a)
b) Broken rays are bisectors c) 90°d) m/NMP + m/PML is 180° since together the angles form a straight angle. So,
6. m/RSZ = 18m/NSZ = 54
7. /VSR > /VST/URT > /URS We could deduce that m/VST = m/URT if we knew /RST > /SRT or /RSV > /SRU.
8. Given: EG bisects /DEFSW bisects /RSTm/DEG = m/RSW
Prove: m/DEF = m/RST
Statement Reason
1. EG bisects /DEF 1. GivenSW bisects /RST
2. /DEG > /GEF 2. Definition of Angle Bisector/RSW > /WST
3. m/DEG = m/GEF 3. Definition of Congruent Anglesm/RSW = m/WST
4. m/DEG + m/RSW = m/GEF + m/WST 4. Addition Property for Equality ( 3 + 3)
5. m/DEG = m/RSW 5. Given
6. m/DEG + m/DEG = m/DEG + m/WST 6. Substitution (5 and 3 into 4)
7. m/DEG + m/DEG + -m/DEG = 7. Addition Property for Equalitym/DEG + m/WST + -m/DEG
8. m/DEG + m/DEG + -m/DEG = 8. Commutative Property of Additionm/DEG + -m/DEG + m/WST
9. m/DEG + 0 = 0 + m/WST 9. Property of Additive Inverse (a + -a = 0)
10. m/DEG = m/WST 10. Identity Property of Addition (a + 0 = a)
11. m/DEG + m/GEF = m/DEF 11. Postulate 7– Protractor – Fourth Assumption – Anglem/RSW + m/WST = m/RST Addition Assumption
12. m/DEG + m/DEG = m/DEF 12. Substitution (3, 5, 10 into 11)m/DEG + m/DEG = m/RST
13. m/DEF = m/RST 13. Substitution (12 into 12)
30 Unit III – Fundamental Theorems
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280 90 m NMP + m PML = 1i i i∠ ∠ =
7. m/RVQ + m/QVT + m/TVP + m/PVS = 7. Substitution (6 into 5)m/RVS
8. m/RVQ + m/RVQ + m/SVP + m/PVS = 8. Substitution (3 into 7)m/RVS
9. m/RVQ + m/RVQ + m/PVS + m/PVS = 180 9. Substitution
10. (1 + 1) • m/RVQ + (1 + 1) • m/PVS = 180 10. Distributive Property of Multiplication over Addition
11. 2 • m/RVQ + 2 • m/PVS = 180 11. Substitution (1 + 1 = 2)
12. 2(m/RVQ + m/PVS) = 180 12. Distributive Property of Multiplication over Addition
13. 13. Multiplication Property for Equality
14. 14. Multiplicative Inverse Property
15. 15. Identity Property of Multiplication
16. m/RVQ + m/PVS = 90 16. Substitution of ( )
17. /RVQ and /PVS are complementary angles 17. Definition of Complementary Angles – Two angles, the sum of
whose measures is 90
11. Given: /APF > /CPDPE bisects /DPF
Prove: PE AC
Statement Reason
1. /APC is a straight angle with a measure of 180° 1. Definition of Straight Angle – An angle whose sides are oppositerays giving a measure of 180°
2. m/APE + m/EPC = m/APC 2. Postulate 7 – Protractor – Fourth Assumption – Angle AdditionAssumption
3. m/APE + m/EPC = 180° 3. Substitution (1 into 2)
4. m/APD + m/DPE = m/APE 4. Postulate 7 – Protractor – Fourth Assumption – Angle Additionm/EPF + m/FPC = m/EPC Assumption
5. m/APD + m/DPE + m/EPF + m/FPC = 180 5. Substitution (4 into 3)
6. /APF > /CPD 6. Given
7. m/APF = m/CPD 7. Definition of Congruent Angles
8. m/APD + m/DPF = m/APF 8. Postulate 7 – Protractor – Fourth Assumption – Angle Additionm/CPF + m/FPD = m/CPD Assumption
9. m/APD + m/DPF = m/CPF + m/FPD 9. Substitution (8 into 7)
10. m/APD + m/DPF + -m/DPF = m/CPF 10. Addition Property for Equality+ m/DPF + -m/DPF
11. m/APD + 0 = m/CPF + 0 11. Additive Inverse Property (a + -a = 0)
*12. m/APD = m/CPF 12. Identity Property for Addition (a + 0 = a)
13. PE bisects /DPF 13. Given
14. /DPE > /EPF 14. Definition of Angle Bisector
*15. m/DPE = m/EPF 15. Definition of Congruent Angles
16. m/APD + m/DPE + m/DPE + m/APD = 180 16. Substitution (15 and 12 into 5)
32 Unit III – Fundamental Theorems
1
2
1
2 2 m RVQ + m PVS 180i i∠ ∠( ) =
1 m RVQ + m PVS 180i i∠ ∠( ) =
1
2
m RVQ + m PVS 180∠ ∠ =
1
2i
1
2 180 = 90i
34 Unit III – Fundamental Theorems34 Unit III – Fundamental Theorems
Unit III – Fundamental TheoremsPart F — Theorems About Angles - Part 2 (Two Angles)
p. 256 – Lesson 1 — Theorem 9: If two adjacent acute angles have their exterior sides in perpendicular lines, then the two angles are complementary.
1. a) Theorem 9 – If two adjacent acute angles have their exterior sides in perpendicular lines, then the two angles are complementary.
b)
c) Given: /AOB and /BOC adjacent acute angles with d) Prove: /AOB and /BOC are complementaryexterior sides that are in perpendicular lines, ,and m.
e)
1. , m 1. Given
2. /AOC is a right angle 2. Definition of Perpendicular Lines
3. m/AOC = 90 3. Definition of Right Angle
4. m/AOB + m/BOC = m/AOC 4. Postulate 7 - (Protractor) Angle-Addition Assumption.
5. m/AOB + m/BOC = 90 5. Substitution of Equals (3 into 4)
6. /AOB and /BOC are complementary angles. 6. Definition of Complementary Angles
2. angle xcomplement 90 – xsupplement 180 – x
The angle measures 30O
3. Given: RU RS; TU TS and m/URT = m/UTRProve: m/RTS = m/TRS
1. RU RS; TU TS 1. Given
2. /URT and /TRS are complementary 2. Theorem 9
3. /UTR and /RTS are complementary 3. Theorem 9
4. m/URT+ m/TRS = 90 4. Definition of Complementary Angles
5. m/UTR + m/RTS = 90 5. Definition of Complementary Angles
6. m/URT + m/TRS = m/UTR + m/RTS 6. Substitution of Equals (3 into 3)
7. m/URT = m/UTR 7. Given
8. m/URT + m/TRS – m/URT = m/UTR 8. Subtraction Property for Equality
+ m/RTS – m/UTR
9. m/URT + m/TRS + —m/URT = m/UTR 9. Definition of Subtraction
+ m/RTS + —m/UTR
10. m/URT + —m/URT + m/TRS = m/UTR 10. Commutative Property of Addition
+ —m/UTR + m/RTS
11. 0 + m/TRS = 0 + m/RTS 11. Additive Inverse Property
12. m/TRS = m/RTS 12. Identity Property of Addition
13. m/RTS = m/TRS 13. Symmetric Property of Equality
>
> >
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902
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530
− = −( )− = −
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=
x x
x x
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x
xx
35
4. Given: OA OC; OB ODProve: /DOC > /AOB
1. OA OC; OB OD 1. Given
2. /AOB and /BOC are complementary 2. Theorem 9
3. /DOC and /BOC are complementary 3. Theorem 9
4. m/AOB + m/BOC = 90 4. Definition of Complementary Angles
5. m/DOC + m/BOC = 90 5. Definition of Complementary Angles
6. m/AOB + m/BOC = m/DOC + m/BOC 6. Substitution of Equals (5 into 4)
7. m/BOC = m/BOC 7. Reflexive Property of Equality
8. m/AOB + m/BOC – m/BOC = m/DOC 8. Subtraction Property for Equality
+ m/BOC – m/BOC
9. m/AOB + m/BOC + —m/BOC = m/DOC 9. Definition of Subtraction
+ m/BOC + —m/BOC
10. m/AOB + 0 = m/DOC + 0 10. Additive Inverse Property
11. m/AOB = m/DOC 11. Identity Property of Addition
12. m/DOC = m/AOB 12. Symmetric Property of Equality
13. /DOC > /AOB 13. Definition of Angle Congruence
5. Given: ST SR<XYZ is a complement of <RSV
Prove: /XYZ > /VST
1. ST SR 1. Given
2. /RSV and /VST are complementary angles 2. Theorem 9
3. /XYZ is a complement of /RSV 3. Given
4. m/RSV + m/VST = 90 4. Definition of Complementary Angles
5. m/XYZ + m/RSV = 90 5. Definition of Complementary Angles
6. m/RSV + m/VST = m/XYZ + m/RSV 6. Substitution of Equals (3 into 3)
7. m/RSV = m/RSV 7. Reflexive Property of Equality
8. m/RSV + m/VST – m/RSV = m/XYZ 8. Subtraction Property for Equality
+ m/RSV – m/RSV
9. m/RSV + m/VST + —m/RSV = m/XYZ 9. Definition of Subtraction
+ m/RSV + —m/RSV
10. m/VST + —m/RSV + m/RSV = m/XYZ 10. Commutative Property of Addition
+ —m/RSV + m/RSV
11. m/VST + 0 = m/XYZ + 0 11. Additive Inverse Property
12. m/VST = m/XYZ 12. Identity Property of Addition
13. m/XYZ = m/VST 13. Symmetric Property of Equality
14. /XYZ > /VST 14. Definition of Congruent Angles
> >
> >
>
>
35Part F – Theorems About Angles – Part 2 (Two Angles)
36 Unit III – Fundamental Theorems
6. Given: /2 > /3/1 and /3 are complementary
Prove: nRUT is a right triangle
1. /1 and /3 are complementary 1. Given
2. m/1 + m/3 = 90 2. Definition of Complementary Angles
3. /2 > /3 3. Given
4. m/2 = m/3 4. Definition of Congruent Angles
5. m/1 + m/2 = 90 5. Substitution of Equals
6. m/1 + m/2 = m/RUT 6. Postulate 7 (Protractor) Angle-Addition Assumption
7. 90 = m/RUT 7. Substitution of Equals
8. /RUT is a right angle 8. Definition of Right Angle
9. nRUT is a right triangle 9. Definition of Right Triangle
7. ORm/2 + m/3 = 90so, m/2 = 90 – m/3
m/1 + m/2 = 180so, m/1 + 90 – m/3 = 180m/1 – m/3 = 90 (substitution)
8. a) If two angles are complementary, then the angles are adjacent acute angles with their exterior sides in perpendicular lines.b) yesc) nod) false
Unit III – Fundamental TheoremsPart F — Theorems About Angles - Part 2 (Two Angles)
p. 258 – Lesson 2 — Theorem 10: If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary.
1. a) Theorem 10 – If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary.
b)
c) Given: /ADC and /COB are adjacent acute angles with d) Prove: /AOC and /COB are supplementaryexterior sides that are opposite rays.
e)
1. /AOC and /COB are adjacent angles with 1. Givenexterior sides that are opposite rays.
2. OC lies between OA and OB 2. Definition of Adjacent Angles
3. m/AOC + m/COB = m/AOB 3. Postulate 7 (Protractor) Angle-Addition Assumption
4. /AOB is a straight angle 4. Definition of Straight Angle - An Angle whose sides are opposite rays
5. m/AOB = 180 5. Definition of Straight Angle - An Angle whose sides are opposite rays
6. m/AOC + /COB = 180 6. Substitution of Equals
7. /AOC and /COB are supplementary 7. Definition of Supplementary Angles
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3737Part F – Theorems About Angles – Part 2 (Two Angles)
2. a) Theorem 10 – Special Case – If one angle of a linear pair is a right angle, then the other angle is a right angle
b)
c) Given: /AOC and /COB are a linear pair d) Prove: /COB is a right angle/AOC is a right angle
e)
1. /AOC and /COB are a linear pair 1. Given
2. /AOC and /COB are adjacent angles with exterior 2. Definition of Linear Pairsides opposite rays
3. /AOC and /COB are supplementary 3. Theorem 10
4. m/AOC + m/COB = 180 4. Definition of Supplementary Angles
5. m/AOC is a right angle 5. Given
6. m/AOC = 90 6. Definition of Right Angle
7. 90 + m/COB = 180 7. Substitution of Equals
8. 90 + m/COB + –90 = 180 + –90 8. Addition Property for Equality
9. m/COB + 90 + –90 = 180 + –90 9. Commutative Property of Addition
10. m/COB + 0 = 180 + –90 10. Additive Inverse Property
11. m/COB = 180 + –90 11. Identity Property of Addition
12. m/COB = 90 12. Substitution of Equals
13. /COB is a right angle 13. Definition of Right Angle
3. a) 180 – 30 = 150b) 180 – 90 = 90c) 180 – 178 = 2
4. The supplement of an acute angle is an obtuse angle.The supplement of an obtuse angle is an acute angle.
5. 180 – n
6. The supplement is 90O more than the complement.
7.
8.
x is the measure of the angley is the measure of the supplementz is the measure of the complement
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x + y = 180
x + x + 20 = 180
2x + 20 = 180
2x = 160
x = 80
y = x + 20 = 1100
x + y = 180
x + z = 90
y = 3z
180 - x = 3 90 - x
180 - x = 270 - 3x
2
( )
xx = 90
x = 45
38 Unit III – Fundamental Theorems
9. Given: Two angles are supplementary and congruent Prove: Each angle is a right angle
Case #1Given: /AOC and /COB are supplementary Prove: /AOC and /COB are right angles
/AOC > /COB
1. /AOC and /COB are supplementary 1. Given
2. m/AOC + m/COB = 180 2. Definition of Supplementary Angles
3. /AOC > /COB 3. Given
4. m/AOC = m/COB 4. Definition of Congruent Angle
5. m/AOC + /AOC = 180 5. Substitution of Equals
6. (1 + 1) m/AOC = 180 6. Distributive Property of Multiplication Over Addition
7. 2 • m/AOC = 180 7. Substitution of Equals
8. 1/2 • 2m/AOC = 1/2 • 180 8. Multiplication Property for Equality
9. 1 • m/AOC = 1/2 • 180 9. Multiplicative Inverse Property
10. m/AOC = 1/2 • 180 10. Identity Property of Multiplication
11. m/AOC = 90 11. Substitution of Equals
12. /AOC is a right angle 12. Definition of Right Angle
13. /AOC and /COB are a linear pair 13. Definition of Linear Pair - (As observed in our case 1 diagram)
14. /COB is a right angle 14. Corollary to Theorem 10 - If one angle of a linear pair is a rightangle, then the other angle is a right angle.
Case #2Given: /AOC and /BPD are supplementary Prove: /AOC and /BPD are right angles
/AOC > /BPD
1. /AOC and /BPD are supplementary 1. Given
2. m/AOC + m/BPD = 180 2. Definition of Supplementary Angles
3. /AOC > /BPD 3. Given
4. m/AOC = m/BPD 4. Definition of Congruent Angle
5. m/AOC + /AOC = 180 or m/BPD + /BPD = 180 5. Substitution of Equals
6. (1 + 1) m/AOC = 180 or (1 + 1) m/BPD = 180 6. Distributive Property of Multiplication Over Addition
7. 2 • m/AOC = 180 or 2 • m/BPD = 180 7. Substitution of Equals
8. 1/2 • 2m/AOC = 1/2 • 180 or 1/2 • 2m/BPD = 1/2 • 180 8. Multiplication Property for Equality
9. 1 • m/AOC = 1/2 • 180 or 1 • m/BPD = 1/2 • 180 9. Multiplicative Inverse Property
10. m/AOC = 1/2 • 180 or m/BPD = 1/2 • 180 10. Identity Property of Multiplication
11. m/AOC = 90 or m/BPD = 90 11. Substitution of Equals
12. /AOC is a right angle or /BPD is a right angle 12. Definition of Right Angle
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3939Part F – Theorems About Angles – Part 2 (Two Angles)
10. Given: Two lines are perpendicular Prove: The angles formed are congruent, adjacent angles
Given: AC BD Prove: /AOB > /BOC; /BOC > /COD/COD > /DOA; /DOA > /AOB
1. AC BD 1. Given
2. /AOB is a right angle 2. Definition of Perpendicular Lines
3. /AOB and /BOC for a linear pair 3. Definition of Linear Pair/BOC and /COD for a linear pair/COD and /DOA for a linear pair/DOA and /AOB for a linear pair
4. /AOB and /BOC are adjacent angles 4. Definition of Linear Pair - ray between exterior sides and exterior /BOC and /COD are adjacent angles sides opposite rays./COD and /DOA are adjacent angles/DOA and /AOB are adjacent angles
5. /BOC, /COD, and /DOA are right angles 5. Corollary to Theorem 10 - If one angle of a linear pair is a right angle, then the other angle is a right angle.
6. m/AOB = 90, m/BOC = 90, m/COD = 90, 6. Definition of Right Anglem/DOA = 90
7. m/AOB = m/BOC = m/COD = m/DOA 7. Substitution
8. /AOB > /BOC > /COD > /DOA 8. Definition of Congruent Angles
11. Given: /1 and /2 are a linear pair/1 > /2
Prove: Line m Line n
1. /1 and /2 are a linear pair 1. Given
2. /1 and /2 are adjacent angles whose exterior 2. Definition of Linear Pairsides are opposite rays
3. /1 and /2 are supplementary 3. Theorem 10
4. m/1+ m/2 = 180 4. Definition of Supplementary Angles
5. /1 > /2 5. Given
6. m/1 = m/2 6. Definition of Congruent Angles
7. m/2 + m/2 = 180 7. Substitution of Equals
8. (1 + 1) m/2 = 180 8. Distributive Property of Multiplication over Addition
9. 2 • m/2 = 180 9. Substitution of Equals
10. 1/2 • 2m/2 = 1/2 • 180 10. Multiplication of Equality
11. 1 • m/2 = 1/2 • 180 11. Multiplicative Inverse
12. m/2 = 1/2 • 180 12. Identity of Multiplication
13. m/2 = 90 13. Multiplication Fact or Substitution
14. /2 is a right angle 14. Definition of Right Angle
15. m n 15. Definition of Perpendicular Lines
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40 Unit III – Fundamental Theorems
12. If two congruent angles form a linear pair, then the intersecting lines forming the angles are perpendicular.
13. a) If two angles are supplementary, then the two angles are adjacent angles with exterior sides in opposite rays.b) Noc) False
Unit III – Fundamental TheoremsPart F — Theorems About Angles - Part 2 (Two Angles)
p. 261 – Lesson 3 — Theorem 11: If you have right angles, then those right angles are congruent.
1. a) Theorem 11 – If you have right angles, then those right angles are congruent.
b)
c) Given: /ABC and /XYZ are right angles d) Prove: /ABC > /XYZ
e)
1. /ABC is a right angle 1. Given
2. /XYZ is a right angle 2. Given
3. m/ABC = 90 3. Definition of Right Angle
4. m/XYZ = 90 4. Definition of Right Angle
5. m/ABC = m/XYZ 5. Substitution of Equals
6. /ABC > /XYZ 6. Definition of Congruence
2. /AQC, /CQE, /EQG, /GQA, /BQD, /DQF, /FQH, /HQB
3.
1. m n 1. Given
2. /AEB, /BEC, /CED, and /DEA are right angles 2. Definition of Perpendicular
3. /AEB > /BEC > /CED > /DEA 3. Theorem 11
4.
1. /BOA is a right angle; /BOC is a right angle 1. Given
2. /BOA > /BOC 2. Theorem 11
3. m/BOA = m/BOC 3. Definition of Congruent Angles
4. /BOE > /BOD 4. Given
5. m/BOE = m/BOD 5. Definition of Congruent Angles
6. m/BOE = m/BOA + m/AOE 6. Postulate 7 (Protractor) - Angle-Addition Assumption
m/BOD = m/BOC + m/COD
7. m/BOA + m/AOE = m/BOC + m/COD 7. Substitution of Equals
8. m/BOA + m/AOE –m/BOA = m/BOC 8. Subtraction Property for Equality
+ m/COD – m/BOC
9. m/BOA + m/AOE + –m/BOA = m/BOC 9. Definition of Subtraction (a – a = a + –a)
+ m/COD + – m/BOC
10. m/BOA + –m/BOA + m/AOE = m/BOC 10. Commutative Property of Addition
+ – m/BOC + m/COD
11. 0 + m/AOE = 0 + m/COD 11. Additive Inverse Property (a + –a = 0)
12. m/AOE = m/COD 12. Identity Property for Addition
13. /AOE > /COD 13. Definition of Congruent Angles
>
C
B
A
O
m
12
3
BA
C
O BA
C
O
y=x+20x yx
BA
C
O C
A
O
D
B
P CA
B
O
D
m
C
A
B Z
X
Y
B
A
O
DC
P -1 0 1 2
123
4141Part F – Theorems About Angles – Part 2 (Two Angles)
5.
1. /MON is a right angle 1. Given
2. /RTS is a right angle 2. Given
3. /MON > /RTS 3. Theorem 11
4. m/MON = m/RTS 4. Definition of Congruent Angles
5. m/MON = m/1 + m/2 5. Postulate 7 (Protractor) Angle-Addition Assumption
6. m/RTS = m/3 + m/4 6. Postulate 7 (Protractor) Angle-Addition Assumption
7. m/1 + m/2 = m/3 + m/4 7. Substitution of Equals
6. Not necessarily; Not necessarily; we need to know that one pair, either /1 and /3 or /2 and /4, is congruent; 45 degrees
7.
1. AB CD; XC CD; YD CD 1. Given
2. /ABC is a right angle; /ABD is a right angle 2. Definition of Perpendicular
/CDY is a right angle; /DCX is a right angle
3. /ABC > /ABD > /CDY > /DCX 3. Theorem 11
4. m/ABC = m/ABD = m/CDY = m/DCX 4. Definition of Congruent Angles
5. m/ABC = m/1 + m/3 5. Postulate 7 (Protractor) Angle-Addition Assumption
m/ABD = m/2 + m/4
6. m/1 + m/3 = m/CDY 6. Substitution Principle
m/2 + m/4 = m/DCX
8.
1. AC AD; AC CB 1. Given
2. /CAD is a right angle; /ACB is a right angle 2. Definition of Perpendicular
3. /CAD > /ACB 3. Theorem 11
4. m/CAD = m/ACB 4. Definition of Congruent Angles
5. /1 > /2 5. Given
6. m/1 = m/2 6. Definition of Congruent Angles
7. m/CAD + m/1 = m/ACB + m/2 7. Addition Property for Equality
8. m/CAD + m/1 = m/DAB 8. Postulate 7 (Protractor) Angle-Addition Assumption
m/ACD + m/2 = m/BCD
9. m/DAB = m/BCD 9. Substitution Principle
10. /DAB > /BCD 10. Definition of Congruent Angles
> > >
> >
42 Unit III – Fundamental Theorems
9.
1. PQ MN 1. Given
2. /MQP is a right angle; /NQP is a right angle 2. Definition of Perpendicular
3. /MQP > /NQP 3. Theorem 11
4. m/MQP = m/NQP 4. Definition of Congruent Angles
5. RQ bisects /MQP; SQ bisects /NQP 5. Given
6. /MQR > /RQP; /NQS > /SQP 6. Definition of Angle Bisector
7. m/MQR = m/RQP; m/NQS = m/SQP 7. Definition of Congruent Angles
8. m/MQR + m/RQP = m/MQP 8. Postulate 7 (Protractor) Angle-Addition Assumption
m/NQS + m/SQP = m/NQP
9. m/MQR + m/RQP = m/NQS + m/SQP 9. Substitution of Equals
10. m/RQP + m/RQP = m/SQP + m/SQP 10. Substitution of Equals
11. (1 + 1) m/RQP = (1 + 1) m/SQP 11. Distributive Property of Multiplication over Addition
12. 2m/RQP = 2m/SQP 12. Substitution of Equals (1 + 1 = 2)
13. 1/2 • 2m/RQP = 1/2 • 2m/SQP 13. Multiplication Property for Equality
14. 1 • m/RQP = 1 • m/SQP 14. Multiplicative Inverse Property
15. m/RQP = m/SQP 15. Identity Property for Multiplication
16. m/SQP = m/RQP 16. Symmetric Property for Equality
17. m/MQP + m/PQS = m/NQP + m/PQR 17. Addition Property for Equations
18. m/MQP + m/PQS = m/MQS 18. Postulate 7 (Protractor) Angle-Addition Assumption
m/NQP + m/PQR = m/NQR
19. m/MQS = m/NQR 19. Substitution of Equals
20. /MQS > /NQR 20. Definition of Congruent Angles
10.
1. /EQF is a straight angle 1. Definition of Straight Angle - An angle whose sides are opposite rays...
2. m/EQF = 180 2. Definition of Straight Angle - ...giving a measure of 180O
3. m/EQF = m/3 + m/BQF; 3. Postulate 7 (Protractor) Angle-Addition Assumption
m/BQF = m/BQD + m/1
4. m/EQF = m/3 + m/BQD + m/1 4. Substitution of Equals
5. 180 = m/3 + m/BQD + m/1 5. Substitution of Equals
6. AB CD at point Q 6. Given
7. /BQD is a right angle 7. Definition of Perpendicular
8. m/BQD = 90 8. Definition of Right Angle
9. 180 = m/3 + 90 + m/1 9. Substitution of Equals
10. 180 + –90 = m/3 + 90 + m/1 + –90 10. Addition Property for Equals
11. 90 = m/3 + 90 + m/1 + –90 11. Substitution of Equals
12. 90 = m/3 + m/1 + 90 + –90 12. Commutative Property of Addition
13. 90 = m/3 + m/1 + 0 13. Additive Inverse Property
14. 90 = m/3 + m/1 14. Identity Property for Addition
15. /1 and /3 are complementary 15. Definition of Complementary Angles
>
>
4343Part F – Theorems About Angles – Part 2 (Two Angles)
Unit III – Fundamental TheoremsPart F — Theorems About Angles - Part 2 (Two Angles)
p. 264 – Lesson 4 — Theorem 12: If you have straight angles, then those straight angles are congruent.
1. a) Theorem 12 – If you have straight angles, then those straight angles are congruent.
b)
c) Given: /AOB and /CPD are straight angles d) Prove: /AOB > /CPD
e)
1. /AOB and /CPD are straight angles 1. Given
2. m/AOB = 180; m/CPD 2. Definition of Straight Angle
3. m/AOB = m/CPD 3. Substitution of Equals
4. /AOB > /AOB 4. Definition of Congruent Angles
2. m/AOB = m/AOF + m/FOBm/AOB = m/AOD + m/DOBm/AOB = m/AOC + m/COBm/AOB = m/AOE + m/EOB
m/COD = m/COA + m/AODm/COD = m/COF + m/FODm/COD = m/COE + m/EODm/COD = m/COB + m/BOD
m/EOF = m/EOC + m/COFm/EOF = m/EOA + m/AOFm/EOF = m/EOB + m/BOFm/EOF = m/EOD + m/DOF
3.
1. /MON is a straight angle; /RTS is a straight angle 1. Given
2. /MON > /RTS 2. Theorem 12
3. m/MON = m/RTS 3. Definition of Congruent Angles
4. m/MON = m/1 + m/2; m/RTS = m/3 + m/4 4. Postulate 7 (Protractor) - Angle-Addition Assumption
5. m/1 + m/2 = m/3 + m/4 5. Substitution of Equals
6. m/1 > m/3 6. Given
7. m/1 = m/3 7. Definition of Congruent Angles
8. m/1 + m/2 –m/1 = m/3 + m/4 – m/3 8. Subtraction Property for Equations
9. m/1 + m/2 + –m/1 = m/3 + m/4 + –m/3 9. Definition of Subtraction
10. m/1 + –m/1 + m/2 = m/3 + –m/3 + m/4 10. Commutative Property of Addition
11. 0 + m/2 = 0 + m/4 11. Additive Inverse Property
12. m/2 = m/4 12. Identity Property of Addition
13. /2 > /4 13. Definition of Congruent Angles
C
B
A
O
m
12
3
BA
C
O BA
C
O
y=x+20x yx
BA
C
O C
A
O
D
B
P CA
B
O
D
m
C
A
B Z
X
Y
B
A
O
DC
P -1 0 1 2
123
44 Unit III – Fundamental Theorems
4.
1. /GHI is a straight angle; /XYZ is a straight angle 1. Given
2. /GHI > /XYZ 2. Theorem 12
3. m/GHI = m/XYZ 3. Definition of Congruent Angles
4. m/GHI = m/1 + m/2; m/XYZ = m/3 + m/4 4. Postulate 7 (Protractor) - Angle-Addition Assumption
5. m/1 + m/2 = m/3 + m/4 5. Substitution of Equals
5. Not necessarily; Not necessarily; we need to know that one pair, either /1 and /3 or /2 and /4, is congruent; 90 degrees.
6.
1. /AQB is a straight angle; /DQC is a straight angle 1. Given
2. /AQB > /DQC 2. Theorem 12
3. m/AQB = m/DQC 3. Definition of Congruent Angles
4. m/DQC = m/3 + m/2; m/AQB = m/2 + m/3 4. Postulate 7 (Protractor) - Angle-Addition Assumption
5. m/1 + m/2 = m/2 + m/3 5. Substitution of Equals
6. m/1 + m/2 – m/2 = m/2 + m/3 – m/2 6. Subtraction Property for Equations
7. m/1 + m/2 + –m/2 = m/2 + m/3 + –m/2 7. Definition of Subtraction
8. m/1 + m/2 + –m/2 = m/3 + m/2 + –m/2 8. Commutative Property of Addition
9. m/1 + 0 = m/3 + 0 9. Additive Inverse Property
10. m/1 = m/3 10. Identity Property for Addition
11. /1 > /3 11. Definition of Congruent Angles
7.
1. /MRV is a straight angle; /MNB is a straight angle 1. Given
2. /MRV > /MNB 2. Theorem 12
3. m/MRV = m/MNB 3. Definition of Congruent Angles
4. m/MRV = m/3 + m/TRV; m/MNB = m/2 + m/SNB 4. Postulate 7 (Protractor) - Angle-Addition Assumption
5. m/2 + m/SNB = m/3 + m/TRV 5. Substitution of Equals
6. /2 > /3 6. Given
7. m/2 = m/3 7. Definition of Congruent Angles
8. m/2 + m/SNB – m/2 = m/3 + m/TRV – m/3 8. Subtraction Property for Equations
9. m/2 + m/SNB + –m/2 = m/3 + m/TRV + –m/3 9. Definition of Subtraction
10. m/SNB + m/2 + –m/2 = m/TRV + m/3 + –m/3 10. Commutative Property of Addition
11. m/SNB + 0 = m/TRV + 0 11. Additive Inverse Property
12. m/SNB = m/TRV 12. Identity Property for Addition
13. /RNT is a straight angle; /NRS is a straight angle 13. Given
14. /RNT > /NRS 14. Theorem 12
15. m/RNT = m/NRS 15. Definition of Congruent Angles
16. m/RNT = m/4 + m/SNB; m/NRS = m/5 + m/TRV 16. Postulate 7 (Protractor) - Angle-Addition Assumption
17. m/4 + m/SNB = m/5 + m/TRV 17. Substitution of Equals
18. m/4 + m/SNB – m/SNB = m/5 + m/TRV – m/TRV 18. Subtraction Property for Equations
19. m/4 + m/SND + –m/SNB = m/5 + m/TRV + –m/TRV 19. Definition of Subtraction
20. m/4 + 0 = m/5 + 0 20. Additive Inverse Property
21. m/4 = m/5 21. Identity Property for Addition
22. /4 > /5 22. Definition of Congruent Angles
4545Part F – Theorems About Angles – Part 2 (Two Angles)
8.
1. /ASC and /BRC are straight angles 1. Given
2. /ASC > /BRC 2. Theorem 12
3. m/ASC = m/BRC 3. Definition of Congruent Angles
4. m/ASC = m/ASB + m/BSC 4. Postulate 7 (Protractor) - Angle-Addition Assumption
m/BRC = m/BRA + m/ARC
5. m/ASB + m/BSC = m/BRA + m/ARC 5. Substitution of Equals
6. /ASB > /BRA 6. Given
7. m/ASB = m/BRA 7. Definition of Congruent Angles
8. m/ASB + m/BSC – m/ASB = m/BRA + 8. Subtraction Property for Equations
m/ARC – m/BRA
9. m/ASB + m/BSC + –m/ASB = m/BRA + 9. Definition of Subtraction
m/ARC + –m/BRA
10. m/ASB + –m/ASB + m/BSC = m/BRA + 10. Commutative Property of Addition
–m/BRA + m/ARC
11. 0 + m/BSC = 0 + m/ARC 11. Additive Inverse Property
12. m/BSC = m/ARC 12. Identity Property for Addition
13. /BSC > /ARC 13. Definition of Congruent Angles
9.
1. AB XY 1. Given
2. /AXY is a right angle; /BXY is a right angle 2. Definition of Perpendicular
3. /AXY > /BXY 3. Theorem 11
4. m/AXY = m/BXY 4. Definition of Congruent Angles
5. m/AXY = m/1 + m/2; m/BXY = m/3 + m/4 5. Postulate 7 (Protractor) - Angle-Addition Assumption
6. m/1 + m/2 = m/3 + m/4 6. Substitution of Equals
7. /1 > /2 7. Given
8. m/1 = m/2 8. Definition of Congruent Angles
9. XY bisects /WXZ 9. Given
10. /2 > /3 10. Definition of Angles Bisector
11. m/2 = m/3 11. Definition of Congruent Angles
12. m/1 = m/3 12. Transitive Property of Equality
13. m/3 + m/3 = m/3 + m/4 13. Substitution (11 and 12 into 6)
14. m/3 = m/3 14. Reflexive Property of Equality
15. m/3 + m/3 + –m/3 = m/3 + m/4 + –m/3 15. Subtraction Property for Equality
16. m/3 + m/3 + –m/3 = m/3 + m/4 + –m/3 16. Definition of Subtraction
17. m/3 + m/3 + –m/3 = m/3 + –m/3 + m/4 17. Commutative Property of Addition
18. m/3 + 0 = 0 + m/4 18. Additive Inverse Property
19. m/3 = m/4 19. Identity Property for Addition
20. /3 > /4 20. Definition of Congruent Angles
>
46 Unit III – Fundamental Theorems
10. a) Yes, the graph has a definite endpoint (starting point) at x = 2/3 and a definite direct (x > 2/3)
b) straight
11. a) 1. /ABC; /ABD2. Point Y3. Point X4. yes5. yes6. yes7. yes, Point B
b) 1. yes2. yes; /ABC, /ACD, or /BCD3. yes4. 180 degrees5. m/ABC = m/ACD = m/BCD6. /ABC > /ACD > /BCD7. no8. cannot judge
cannot judge
c) no; no
Unit III – Fundamental TheoremsPart G — Theorems About Angles – Part 3 (More Than Two Angles)
p. 268 – Lesson 1 — Theorem 13: If two angles are complementary to the same angles or congruent angles, then they are congruent to each other.
1. a) If two angles are complementary to the same angles or congruent angles, then they are congruent to each other.
b)
c) Given: /1 is the complement of /3 d) Prove: /1 > /2 /2 is the complement of /3
e)
1. /1 is complementary to /3 1. Given
2. m/1 + m/3 = 90 2. Definition of Complementary Angles
3. /2 is complementary to /3 3. Given
4. m/2 + m/3 = 90 4. Definition of Complementary Angles
5. m/1 + m/3 = m/2 + m/3 5. Algebraic Substitution (4 into 2)
6. m/1 + m/3 + –m/3 = m/2 + m/3 + –m/3 6. Addition Property for Equality
7. m/1 + 0 = m/2 + 0 7. Additive Inverse Property (a + –a = 0)
8. m/1 = m/2 8. Identity Property for Addition (a + 0 = a)
9. /1 > /2 9. Definition of Congruent Angles
3x + 7 9
3x + 7 + -7 9 + -7
3x + 0 2
3x 2
13
3x13
2
1 x2
≥≥≥≥
⋅ ≥ ⋅
⋅ ≥33
x23
≥
C
B
A
O
m
12
3
BA
C
O BA
C
O
y=x+20x yx
BA
C
O C
A
O
D
B
P CA
B
O
D
m
C
A
B Z
X
Y
B
A
O
DC
P -1 0 1 2
123
1
3
2 1
2
4
3
1
1
3E
2
D
BC
A
4 24
2 3
1
3E
D
BC
A
1 2 3 4
C
BA
O
m
,
32
1
O B
C
A O B
C
A
x y = x + 20
x y = 3xO B
C
A
O C
B
D
A
O C
A
P D
B
m
,
BC
A
Y Z
X
4747Part G — Theorems About Angles – Part 3 (More Than Two Angles)
2. a) Theorem 13 - Part 2 - If two angles are complementary to congruent angles, then they are congruent to each other.
b)
c) Given: /1 is complementary to /2 d) Prove: /1 > /4 /4 is complementary to /3/2 > /3
e)
1. /1 is complementary to /2 1. Given
/4 is complementary to /3
2. m/1 + m/2 = 90 2. Definition of Complementary Angles
m/4 + m/3 = 90
3. m/1 + m/2 = m/4 + m/3 3. Substitution of Equals (2 into 2)
4. /2 > /3 4. GIven
5. m/2 = m/3 5. Definition of Congruent Angles
6. m/1 + m/2 – m/2 = m/4 + m/3 – m/3 6. Subtraction Property for Equality
7. m/1 + m/2 + –m/2 = m/4 + m/3 + –m/3 7. Definition of Subtraction
8. m/1 + 0 = m/4 + 0 8. Additive Inverse Property
9. m/1 = m/4 9. Identity Property for Addition
10. /1 > /4 10. Definition of Congruent Angles
3. a) 90 – 38 b) 90 – x c) 90 – 2y52
4. The angles are x and 90 – x.
5. a) b)
x = 90 - x
x + x = 90 - x + x
2x = 90 - 0
2x = 90
12
2x =12
90
1 x =
⋅ ⋅
⋅ 445
x = 45
3y + 5 + 2y = 90
5y + 5 = 90
5y + 5 - 5 = 90 - 5
5y + 0 = 85
5y = 85
1
( )
555y =
15
85
1 y = 17
y = 17
⋅ ⋅
⋅
m ACB = 3y + 5
= 3(17) + 5
= 51+ 5
m ACB = 56
m EDF = 2y
= 2(1
∠
∠
∠77)
m EDF = 34∠
(y - 8) + (3y + 2) = 90
4y - 6 = 90
4y - 6 + 6 = 90 + 6
4y - 0 = 96
4y = 996
14
4y =14
96
1 y = 24
y = 24
⋅ ( )
⋅
⋅
m ACB = y - 8
= 24 - 8
m ACB = 16
m DEF = 3y + 2
= 3(24) + 2
= 72
∠
∠
∠
++ 2
m EDF = 74∠
1
3
2 1
2
4
3
1
1
3E
2
D
BC
A
4 24
2 3
1
3E
D
BC
A
1 2 3 4
C
BA
O
m
,
32
1
O B
C
A O B
C
A
x y = x + 20
x y = 3xO B
C
A
O C
B
D
A
O C
A
P D
B
m
,
BC
A
Y Z
X
48 Unit III – Fundamental Theorems
c)
6. The angles are x and 90 – x
7. Theorem 13 - If two angles are complementary to congruent angles, then the two angles are congruent to each other.
8.
1. BD bisects /ABC 1. Given
2. /1 > /2 2. Definition of a Bisector of an Angle
3. /4 is complementary to /1 3. Given
/3 is complementary to /2
4. /4 > /3 4. Theorem 13 - If two angles are complementary to congruent
angles, then the angles are congruent to each other.
9.
1. HP PQ; JQ PQ 1. Given
2. /HPQ is a right angle 2. Definition of Perpendicular
/JQP is a right angle
3. m/HPQ = 90; m/JQP = 90 3. Definition of Right Angle
4. m/HPQ = m/1 + m/2 4. Postulate 7 (Protractor) - Angle-Addition Assumption
m/JQP = m/4 + m/3
5. 90 = m/1 + m/2 5. Substitution of Equals
90 = m/4 + m/3
6. /1 is complementary to /2 6. Definition of Complementary Angles
/4 is complementary to /3
7. /2 > /3 7. Given
8. /1 > /4 8. Theorem 13 - If two angles are complementary to congruent
angles, then the angles are congruent to each other.
y + (6y -1) = 90
y + 6y -1= 90
y + 6y -1- 90 = 90 - 90
y + 6y
2
2
2
2 -- 91= 0
(y - 7)(y +13) = 0
m ACB = y
= 7
m ACB = 49
m EDF = 6y -1
= 6(7) -1
= 42 -1
m
2
2
∠
∠
∠
∠EEDF = 41
y - 7 = 0 or y +13 = 0
y = 7 or y = -13
x =12
90 - x
2 x = 212
90 - x
2x = 1 90 - x
2x = 90 - x
( )
⋅ ⋅ ⋅ ( )⋅ ( )
22x + x = 90 - x + x
3x = 90 - 0
13
3x =13
90
1 x = 30
x = 30
⋅ ⋅
⋅
> >
4949Part G — Theorems About Angles – Part 3 (More Than Two Angles)
10.
1. /JML is a right angle 1. Given
2. m/JML = 90 2. Definition of Right Angles
3. m/JML = m/3 + m/4 3. Postulate 7 (Protractor) - Angle-Addition Assumption
4. 90 = m/3 + m/4 4. Substitution of Equals
5. /3 is complementary to /4 5. Definition of Complementary Angles
6. /3 is complementary to /1 6. Given
7. /1 > /4 7. Theorem 13 - If two angles are complementary to congruent
angles, then the angles are congruent to each other.
11.
1. /ABD is complementary to /BAD 1. Given
/CBE is complementary to /BAD
2. /ABD > /CBE 2. Theorem 13 - If two angles are complementary to the same angle,
then the angles are congruent to each other.
3. m/ABD = m/CBE 3. Definition of Congruent Angles
4. m/DBE = m/EBD 4. Reflexive Property of Equality
5. m/ABD + m/DBE = m/CBE + m/EBD 5. Addition Property for Equality
6. m/ABD + m/DBE = m/ABE 6. Postulate 7 (Protractor) - Angle-Addition Assumption
m/CBE + m/EBD = m/CBD
7. m/ABE = m/CBD 7. Substitution of Equals
8. /ABE > /CBD 8. Definition of Congruent Angles
12.
1. BA CA 1. Given
2. /BAC is a right angle 2. Definition of Perpendicular
3. m/BAC = 90 3. Definition of Right Angle
4. m/1 + m/2 = m/BAC 4. Postulate 7 (Protractor) - Angle-Addition Assumption
5. m/1 + m/2 = 90 5. Substitution of Equals
6. /1 is complementary to /2 6. Definition of Complementary Angles
7. /1 is complementary to /3 7. Given
8. /2 > /3 8. Theorem 13 - If two angles are supplementary to the same angle,
then the angles are congruent to each other.
>
50 Unit III – Fundamental Theorems
Unit III – Fundamental TheoremsPart G — Theorems About Angles – Part 3 (More Than Two Angles)
p. 271 – Lesson 2 — Theorem 14: If two angles are supplementary to the same angle, or congruent angles then they are congruent toeach other.
1. a) Theorem 14 – Part 2 - If two angles are supplementary to the same angle, then they are congruent to each other.
b)
c) Given: /1 is supplementary to /2 d) Prove: /1 > /3/3 is supplementary to /2
e)
1. /1 is supplementary to /2 1. Given
2. m/1 + m/2 = 180 2. Definition of Supplementary Angles
3. /3 is supplementary to /2 3. Given
4. m/3 + m/2 = 180 4. Definition of Supplementary Angles
5. m/1 + m/2 = m/3 +m/2 5. Algebraic Substitution
6. m/2 = m/2 6. Reflexive Property of Equality
7. m/1 + m/2 – m/2 = m/3 + m/2 – m/2 7. Subtraction Property of Equality
8. m/1 + m/2 + –m/2 = m/3 + m/2 + –m/2 8. Definition of Subtraction
9. m/1 + 0 = m/3 + 0 9. Additive Inverse Property
10. m/1 = m/3 10. Identity Property for Addition
11. /1 > /3 11. Definition of Congruent Angles
2. a) Theorem 14 – Part 1 - If two angles are supplementary to congruent angles, then they are congruent to each other.
b)
c) Given: /1 is supplementary to /2 d) Prove: /1 > /4/4 is supplementary to /3/2 > /3
e)
1. /1 is supplementary to /2 1. Given
/4 is supplementary to /3
2. m/1 + m/2 = 180 2. Definition of Supplementary Angles
m/4 + m/3 = 180
3. m/1 + m/2 = m/4 + m/3 3. Substitution of Equals
4. /2 > /3 4. Given
5. m/2 = m/3 5. Definition of Congruent Angles
6. m/1 + m/2 – m/2 = m/4 + m/3 – m/3 6. Subtraction Property of Equality
7. m/1 + m/2 + –m/2 = m/4 + m/3 + –m/3 7. Definition of Subtraction
8. m/1 + 0 = m/4 + 0 8. Additive Inverse Property
9. m/1 = m/4 9. Identity Property for Addition
11. /1 > /4 10. Definition of Congruent Angles
1
3
2 1
2
4
3
1
1
3E
2
D
BC
A
4 24
2 3
1
3E
D
BC
A
1 2 3 4
C
BA
O
m
,
32
1
O B
C
A O B
C
A
x y = x + 20
x y = 3xO B
C
A
O C
B
D
A
O C
A
P D
B
m
,
BC
A
Y Z
X
1
3
2 1
2
4
3
1
1
3E
2
D
BC
A
4 24
2 3
1
3E
D
BC
A
1 2 3 4
C
BA
O
m
,
32
1
O B
C
A O B
C
A
x y = x + 20
x y = 3xO B
C
A
O C
B
D
A
O C
A
P D
B
m
,
BC
A
Y Z
X
5151Part G — Theorems About Angles – Part 3 (More Than Two Angles)
3. a) 180 – 42 b) 180 – (x – 3) c) 180 – x2
138 180 – x + 3183 – x
4. The angles are x and 180 – x.
5. a) b)
c)
6. The angles x and 180 – x.
7. Theorem 14 - If two angles are supplements of congruent angles, then they are congruent to each other.
x = 180 - x
x + x = 180 - x + x
2x = 180 - 0
2x = 180
12
2x =12
18⋅ ⋅ 00
1 x = 90
x = 90
⋅
2x + x -15 = 180
2x + x -15 = 180
3x -15 = 180
3x -15 +15 = 1
( )
880 +15
3x - 0 = 195
3x = 195
13
3x =13
195
1 x = 65
x = 65
⋅ ⋅
⋅
m ACB = 2x
= 2 65
m ACB = 130
m EDF = x -15
= 65 -15
m ED
∠( )
∠
∠
∠ FF = 50
x + 12x - 9 = 180
x +12x - 9 = 180
x +12x - 9 -180 = 180 -
2
2
2
( )
1180
x +12x -189 = 0
x - 9 x + 21 = 0
2
( )( )
m ACB = x
= 9
m ACB = 81
m DEF = 12x - 9
= 12 9 - 9
= 108 -
2
2
∠
∠
∠( )
99
m DEF = 99∠
x +16 + 2x -16 = 180
x +16 + 2x -16 = 180
3x = 180
13
3x
( ) ( )
⋅ ==13
180
1 x = 60
x = 60
⋅
⋅
m ACB = x +16
= 60 +16
m ACB = 76
m EDF = 2x -16
= 2 60 -1
∠
∠
∠( ) 66
= 120 -16
m EDF = 104∠
x = 2 180 - x
x = 360 - 2x
x + 2x = 360 - 2x + 2x
3x = 360 + 0
3x =
( )
3360
13
3x =13
360
1 x = 120
x = 120
⋅ ⋅
⋅
52 Unit III – Fundamental Theorems
8.
1. /ABC and /DCB are straight angles 1. Definition of Straight Angle
2. m/ABC = 180; m/DCB = 180 2. Definition of Straight Angle
3. m/ABC = m/1 + m/2 3. Postulate 7 (Protractor) - Angle-Addition Assumption
m/DCB = m/3 + m/4
4. 180 = m/1 + m/2; 180 = m/3 + m/4 4. Substitution of Equals
5. /1 is the supplement of /2 5. Definition of Supplementary Angles
/3 is the supplement of /4
6. /2 > /3 6. Given
7. /1 > /4 7. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
9.
1. /BFE > /ECA; /CFD > /ECA 1. Given
2. m/BFE = m/ECA; m/CFD = m/ECA 2. Definition of Congruent Angles
3. m/ECA = m/CFD 3. Symmetric Property of Equality
4. m/BFE = m/CFD 4. Transitive Property of Equality
5. /BFE > /CFD 5. Definition of Congruent Angles
6. m/BFE + m/EFC = m/BFC 6. Postulate 7 (Protractor) - Angle-Addition Assumption
m/CFD + m/DFB = m/CFB
7. /BFC is a straight angle 7. Definition of Straight Angle
8. m/BFC = 180; m/BFC = 180 8. Definition of Straight Angle
9. m/BFE + m/EFC = 180 9. Substitution of Equals
m/CED + m/DFB = 180
10. /BFE and /EFC are supplementary angles 10. Definition of Supplementary angles - sum is 180
/CFD and /DFB are supplementary angles
11. /DFB > /EFC 11. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
10.
1. /ACD is a straight angle 1. Definition of Straight Angle
2. m/ACD = 180 2. Definition of Straight Angle
3. m/ACD = m/2 + m/3 3. Postulate 7 (Protractor) - Angle-Addition Assumption
4. 180 = m/2 + m/3 4. Substitution of Equals
5. /2 and /3 are supplementary angles 5. Definition of Supplementary Angles
6. /1 is the supplement of /2 6. Given
7. /1 > /3 7. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
5353Part G — Theorems About Angles – Part 3 (More Than Two Angles)
11.
1. /DBE is a straight angle 1. Definition of Straight Angle
/ECD is a straight angle
2. m/DBE = 180; m/ECD = 180 2. Definition of Straight Angle
3. m/DBE = m/DBA + m/ABE 3. Postulate 7 (Protractor) - Angle-Addition Assumption
m/ECD = m/ECA + m/ACD
4. 180 = m/DBA + m/ABE; 180 = m/ECA + m/ACD 4. Substitution of Equals
5. /DBA and /ABE are supplementary angles 5. Definition of Supplementary Angles
/ECA and /ACD are supplementary angles
6. /ABE > /ACD 6. Given
7. /DBA > /ECA 7. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
8. m/DBA = m/ECA 8. Definition of Congruent Angles
9. /FCA is a straight angle 9. Definition of Straight Angle
10. m/FCA = 180 10. Definition of Straight Angle
11. m/FCA = m/FCE + m/ECA 11. Postulate 7 (Protractor) - Angle-Addition Assumption
12. 180 = m/FCE + m/ECA 12. Substitution of Equals
13. 180 = m/FCE + m/DBA 13. Substitution of Equals
14. /DBA is a supplement of /FCE 14. Definition of Supplementary Angles
12.
1. m/EGB + m/BGF = 180 1. Given
m/EHD + m/BGF = 180
2. /EGB and /BGF are supplementary angles 2. Definition of Supplementary Angles
/EHD and /BGF are supplementary angles
3. /EGB > /EHD 3. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
4. m/EGB = m/EHD 4. Definition of Congruent Angles
Unit III – Fundamental TheoremsPart G — Theorems About Angles – Part 3 (More Than Two Angles)
p. 274 – Lesson 3 — Theorem 15: If two lines intersect, then the vertical angles formed are congruent.
1. a) Theorem 15 – Part 1 - If two lines intersect, then the vertical angles formed are congruent.
b)
c) Given: AB and CD intersect at point E d) Prove: /1 > /3
1
3
2 1
2
4
3
1
1
3E
2
D
BC
A
4 24
2 3
1
3E
D
BC
A
1 2 3 4
C
BA
O
m
,
32
1
O B
C
A O B
C
A
x y = x + 20
x y = 3xO B
C
A
O C
B
D
A
O C
A
P D
B
m
,
BC
A
Y Z
X
54 Unit III – Fundamental Theorems
e)
1. AB and CD intersect at point E 1. Given
2. ED and EC are opposite rays 2. Definition of Opposite Rays
EB and EA are opposite rays
3. /1 is adjacent to /2 3. Definition of Adjacent Angles
/3 is adjacent to /2
4. /1 is supplementary to /2 4. Theorem 10 - If the exterior sides of two adjacent angles are
/3 is supplementary to /2 opposite rays, then the angles are supplementary
5. /1 > /3 5. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
2. a) Theorem 15 – Part 2 - If two lines intersect, then the vertical angles formed are congruent.
b)
c) Given: AB and CD intersect at point E d) Prove: /2 > /4
e)
1. AB and CD intersect at point E 1. Given
2. ED and EC are opposite rays 2. Definition of Opposite Rays
EB and EA are opposite rays
3. /4 is adjacent to /3 3. Definition of Adjacent Angles
/2 is adjacent to /3
4. /4 is supplementary to /3 4. Theorem 10 - If the exterior sides of two adjacent angles are
/2 is supplementary to /3 opposite rays, then the angles are supplementary
5. /4 > /2 5. Theorem 14 - If two angles are supplements of congruent angles,
then they are congruent to each other.
3. a) /3 and /4 are vertical angles; m/3 = 110. Therefore, m/4 = 110
b) /1 and /3 are supplementary
m/1 + m/3 = 180
m/1 + 110 = 180
m/1 = 70
c) /2 and /1 are vertical angles; m/1 = 70. Therefore, m/2 = 70
1
3
2 1
2
4
3
1
1
3E
2
D
BC
A
4 24
2 3
1
3E
D
BC
A
1 2 3 4
C
BA
O
m
,
32
1
O B
C
A O B
C
A
x y = x + 20
x y = 3xO B
C
A
O C
B
D
A
O C
A
P D
B
m
,
BC
A
Y Z
X
5555Part G — Theorems About Angles – Part 3 (More Than Two Angles)
4. a) b) /1 and /2 are vertical angles
m/1 = 60
therefore, m/2 = 60
c) m/3 = 120 (part a)
d) /3 and /4 are vertical angles
m/3 = 120
therefore, m/4 = 120
5. Two angles are congruent and supplementary. Each angle measures 90. Therefore, line m line n.
6. Two angles are congruent and complementary. Each angle measures 45. Therefore, m/4 = 45
7. a) increase by 10 b) decreased by 10 c) decrease by 10
8.
9.
10.
11.
m m
m m
m m
∠ + ∠ =
⋅ ∠ + ∠ =
⋅ ∠ + ∠ =
1 3 180
1
23 3 180
21
23 3 180
∠ + ∠ =⋅ ∠ =
⋅ ∠ = ⋅
⋅ ∠
m m
m
m
m
3 2 3 360
3 3 360
1
33 3
1
3360
1 33 120
3 120
11
23
11
2120
1 60
=∠ =
∠ = ⋅ ∠
∠ = ⋅
∠ =
m
m m
m
m
5 20 4 15
5 4 20 20 4 4 15 20
0 0 35
x x
x x x x
x
x
− = +− − + = − + +
+ = +== 35
5 20 155
4 15 155
25
x m APX
x m BPY
m APB
m XPY
− = = ∠+ = = ∠
∠ =∠ == 25
7 35 3 85
7 3 35 35 3 3 85 35
4 50
1
44
x x
x x x x
x
+ = +− + − = − + −
=
⋅ xx
x
x or
= ⋅
⋅ =
=
1
450
150
425
212
1
2
7 35 122 5
3 85 122 5
57
x m NQT
x m RQM
m NQP
+ = = ∠+ = = ∠
∠ =
.
.
.55
57 5m TQM∠ = .
4 36 64
4 100
1
44
1
4100
1100
425
x
x
x
x
x
= +=
⋅ = ⋅
⋅ =
=
4 180
80
80
64
3
x m QPR
m QPR
m MPU
m UPT
m TPR
+ ∠ =∠ =∠ =∠ =∠ = 66
x x
x x
x x
2
2
6 9
6 9 0
3 3 0
= −
− + =−( ) −( ) =
x or x
x x
− = − == =
3 0 3 0
3 3
x m XPW
x m YPZ
m WPZ
m XPY
2 9
6 9 9
171
171
= = ∠− = = ∠
∠ =∠ =
>
56 Unit III – Fundamental Theorems
12.
1. /1 > /2 1. Given
2. /2 > /4 2. Theorem 15 - If two lines intersect, then the vertical angles
formed are congruent
3. /3 > /4 3. Given
4. m/1 = m/2; m/2 = m/4; m/3 = m/4 4. Definition of Congruent Angles
5. m/4 = m/3 5. Symmetric Property for Equality
6. m/1 = m/3 6. Transitive Property for Equality
7. /1 > /3 7. Definition of Congruent Angles
13.
1. /APR > /NPB 1. Given
2. /NPB > /MPA 2. Theorem 15 - If two lines intersect, then the vertical angles
formed are congruent
3. m/APR = m/NPB 3. Definition of Congruent Angles
m/NPB = m/MPA
4. m/APR = m/MPA 4. Transitive Property of Equality
5. m/MPA = m/APR 5. Symmetric Property of Equality
6. /MPA > /APR 6. Definition of Congruent Angles
14.
1. /MPR > /NPQ 1. Theorem 15 - If two lines intersect, then the vertical angles
formed are congruent
2. m/MPR = m/NPQ 2. Definition of Congruent Angles
3. m/MPR = m/MPA + m/APR 3. Postulate 7 (Protractor) - Angle-Addition Assumption
m/NPQ = m/NPB + m/BPQ
4. m/MPA + m/APR = m/NPB + m/BPQ 4. Substitution of Equals (2 into 2)
5. /APR > /NPB 5. Given
6. m/APR = m/NPB 6. Definition of Congruent Angles
7. m/MPA + m/APR – m/APR = m/NPB + 7. Subtraction Property for Equality
m/BPQ – m/NPB
8. m/MPA + m/APR + –m/APR = m/NPB + 8. Definition of Subtraction
m/BPQ + –m/NPB
9. m/MPA + m/APR + –m/APR =m/BPQ + 9. Commutative Property of Addition
m/NPB + –m/NPB
10. m/MPA + 0 = m/BPQ + 0 10. Additive Inverse Property
11. m/MPA = m/BPQ 11. Identity Property for Addition
12. /MPA > /QPB 12. Definition of Congruent Angles
5757Part G — Theorems About Angles – Part 3 (More Than Two Angles)
15.
1. PA bisects /MPR 1. Given
2. /APR > /APM 2. Definition of Angle Bisector
3. /APR > /BPQ; /APM > /BPN 3. Theorem 15 - If two lines intersect, then the vertical angles
formed are congruent
4. m/APR = m/APM; m/APM = m/BPN 4. Definition of Congruent Angles
5. m/APR = m/BPN 5. Transitive Property of Equality
6. m/APR = m/BPQ 6. Definition of Congruent Angles
7. m/BPN = m/APR 7. Symmetric Property of Equality
8. m/BPN = m/BPQ 8. Transitive Property of Equality
9. /BPN > /BPQ 9. Definition of Congruent Angles
10. PB is between PQ and PN so that 10. Definition of Betweeness for Rays
m/NPB + m/BPQ = m/NPQ
11. PB bisects /NPQ 11. Definition of Angle Bisector
Unit III — Fundamental TheoremsPart H — Theorems about Parallel Lines
p. 277 – Lesson 1 — Postulate 11– Corresponding Angles of Parallel Lines: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
1. Postulate 11 - If two parallel lines are cut by a transversal, then corresponding angles are congruent.
2. a) Yes b) No c) No d) Yes
3. a) ,1 b) ,1 c) ,1 d) ,2 �
,2 ,2 ,3 ,3 �
,3 ,3 ,4 ,4 �
,4 ,5 �
4. a) 1. /2, /3, /6, /7 b) 1. /3, /4, /5, /6
2. /1, /4, /5, /8 2. /1, /2, /7, /8
3. /1, /2, /3, /4 3. /2, /4, /6, /8
4. /5, /6, /7, /8 4. /1, /3, /5, /7
5. a) /2 and /3 b) /1 and /5 c) /3 and /6 d) /2 and /7
/1 and /4 /3 and /7 /4 and /5 /1 and /8�
/8 and /5 /2 and /6
/6 and /7 /4 and /8
6. a) Alternate Interior f) Alternate Exterior
b) Corresponding g) Interior Angles on the
c) Alternate Exterior same side of the transversal
d) Interior Angles on the h) Corresponding
same side of the transversal i) Vertical
e) Alternate Interior
7. a) No Yes No
b) No No No
c) No Yes Yes
d) No No Yes
e) No No Yes
f) No Yes Yes
8. Statement Reason
1. BC EF 1. Given
BA ED
2. /B > /DPC; /DPC > /E 2. Postulate 11 - If two parallel lines are cut by a transversal,m/B 5 m/DPC then corresponding angles are congruent.
3. m/B 5 m/DPC 3. Definition of congruent angles.m/DPC 5 m/E
4. m/B 5 m/E 4. Transitive Property of Equality
5. /B > /E 5. Definition of Congruent Angles
1 2
3 4
5 6
7 8
<1
<2
t
/1 > /5
/3 > /7
/2 > /6
/4 > /8
58 Unit III – Fundamental Theorems
Unit III — Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 282 – Lesson 2 — Theorem 16: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
1. a) Theorem 16 - If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
b)
c) Given: Line , is parallel to line m cut by transversal t.d) Prove: Alternate Interior angles are congruent.
e) Statement Reason
1. , || m, cut by transversal t 1. Given
2. /3 > /7 2. Postulate 11 - If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
3. m/3 = m/7 3. Definition of Congruent Angles
4. /6 > /7 4. Theorem 15 - If two lines intersect, then the vertical angles
formed are congruent.
5. m/6 = m/7 5. Definition of Congruent Angles
6. m/6 = m/3 6. Substitution (3 into 5)
7. /6 > /3 7. Definition of Congruent Angles
2.
Statement Reason
1. , || m ; , and m are cut by transversal t 1. Given
2. /3 and /4 form a Linear Pair 2. Definition of Linear Pair - Two angles which have a common
/5 and /6 form a Linear Pair side (they are adjacent), and whose exterior sides are
opposite rays.
3. /3 and /4 form a Straight Angle 3. Definition of Straight Angle - Sides are opposite rays, giving
/5 and /6 form a Straight Angle a measure of 180 degrees.
4. m/3 + m/4 = 180 4. Definition of Straight Angle.
m/5 + m/6 = 180
5. /3 and /4 are supplementary angles 5. Definition of Supplementary Angles
/5 and /6 are supplementary angles
6. /6 > /3 6. Part I of Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
7. /5 > /4 7. Theorem 14 - If two angles are supplementary to congruent
angles, then they are congruent to each other.
1 2
3 4
5 6
7 8
<
m
t
1 2
3 4
5 6
7 8
<
m
t
60 Unit III – Fundamental Theorems
61Part H – Theorems About Parallel Lines
3. Corollary 16a - If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
c) Given: Line , || m; , and m are cut by a transversal t.d) Prove: Alternate Exterior Angles are Congruent.
e) Statement Reason
1. , || m; , and m are cut by a transversal t. 1. Given
2. /3 > /6 2. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
3. /2 > /3 3. Theorem 15 - If two lines intersect, then the vertical
/6 > /7 angles formed are congruent.
4. m/2 = m/3 4. Definition of Congruent Angles.
m/3 = m/6
m/6 = m/7
5. m/1 = m/2 5. Transitive Property of Equality.
6. /2 > /7 6. Definition of Congruent Angles
4.
Statement Reason
1. , || m ; , and m are cut by transversal t 1. Given
2. /1 and /2 form a Linear Pair 2. Definition of Linear Pair - Two Angles which have a common
/7 and /8 form a Linear Pair side (they are adjacent), and whose exterior sides are
opposite rays.
3. /1 and /2 form a Straight Angle 3. Definition of Straight Angle - Sides are opposite rays, giving
/7 and /8 form a Straight Angle a measure of 180 degrees.
4. m/1 + m/2 = 180 4. Definition of Straight Angle.
m/7 + m/8 = 180
5. /1 and /2 are supplementary angles 5. Definition of Supplementary Angles
/7 and /8 are supplementary angles
6. /2 > /7 6. Part I of Corollary 16a - If two parallel lines are cut by a transversal,
then alternate exterior angles are congruent.
7. /1 > /8 7. Theorem 14 - If two angles are supplementary to congruent angles,
then they are congruent to each other
1 2
3 4
5 6
7 8
<
m
t
1 2
3 4
5 6
7 8
<
m
t
5. m/3 = 100 - 55 = 45
6. 4x - 40 = x + 20 m/ABD = 4x -40
4x - 40 + 40 = x + 20 + 40 = 4(20 - 40
4x = x + 60 = 80 - 40
4x - x = x + 60 - x m/ABD = 40
3x = 60
1/3 • 3 x = 1/3 • 60x = 20
7. a) 75 d) 105
b) 75 e) 105
c) 75 f) 105
8. a) Corresponding Angles b) Alternate Interior Angles
m/1 = m/6 m/4 = m/6
3x + 7 = 5x - 3 8x + 12 = 2x + 54
3x + 7 - 3x = 5x - 3 - 3x 8x - 2x + 12 = 2x + 54 - 2x
7 = 2x - 3 6x + 12 = 54
7 + 3 = 2x - 3 + 3 6x + 12 - 12 + 54 - 12
10 = 2x 6x = 42
1/2 • 10 = 1/2 • 2x 1/6 • 6x = 1/6 • 42
5 = x x = 7
c) Corresponding Angles
m/4 = m/8
x2 + 5x = 9x + 12
x2 + 5x - 9x - 12 = 9x + 12 - 9x - 12
x2 - 4x - 12 = 0
(x - 6) (x + 2) = 0
x - 6 = 0 x + 2 + 0
x = 6 x = -2 This answer creates an angle which measures - 6 degrees Can’t use.
9. Corresponding Angles Congruent
1. /ABE > /ACD /ABE , /EBD, and /DBC form a straight angles.
m/ABE = m/ACD m/ABE + m/EBD + m/DBC = 180
x = 57º x + y + 63 = 180
57 + y + 63 = 180
y + 120 = 180
y + 120 - 120 = 180 -120
y = 60
10. /AED and /ABC are Congruent Corresponding Angles
/AED is a right angle. So, m/AED = 90.
Therefore, m/ABC = 90.
Since m/ABD + m/DBC = m/ABC,
51 + m/DBC = 90
So m/DBC = 90 - 51
= 39 = x
62 Unit III – Fundamental Theorems
63Part H – Theorems About Parallel Lines
10. Continued
/DBC and /BDE are Congruent Alternate Interior Angles
Thus, m/BDE = x = 39
/BDE, /BDC, and /CDF form a straight angle
m/BDE + m/BDC + m/CDF = 180
39 + m/BDC + 68 = 180
m/BDC + 107 = 180
m/BDC + 107 = 180
m/BDC = 73 = z
/EDC and /DCG are Congruent Alternate Interior Angles.
m/EDC = m/EDB + m/BDC
= x + m/BDC
= 39 + 73
m/EDC = 112º
Therefore, m/DCG = 112 = y
11.
/ACD and m/ABE are Congruent Corresponding Angles.
(5x + y) + (5x - y) = 80
10x = 80
1/10 • 10 x = 1/10 • 80x = 8
/BEC and /ECD are Congruent Alternate Interior Angles.
2x + y = 5x - y m/ACE = 5x + y
2x + y + y = 5x - y + y = 5 (8) + 12
2x + 2y = 5x = 40 + 12
2x + 2y = 5x m/ACE = 52
2y = 3x
2y = 3 • 8 m/BEC = 2x + y
2y = 24 2 (8) + 12
1/2 • 2y = 1/2 • 24 = 16 + 12
y = 12 m/BEC = 28
12.
Statement Reason
1. p || q, s || t 1. Given
2. /GHC > /IBD 2. Theorem 16 - If two parallel lines are cut by a transversal,
/IBD > /AEF then alternate interior angles are congruent.
3. m/GHC = m/IBD 3. Definition of Congruent Angles.
m/IBD = m/AEF
4. m/GHC = m/AEF 4. Transitive Property of Equality
5. /GHC > /AEF 5. Definition of Congruent Angles.
64 Unit III – Fundamental Theorems
13. Statement Reason
1. , || m, p || q 1. Given
2. /2 > /4 2. Postulate 11 - If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
3. /4 > /9 3. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
4. m/2 = m/4 4. Definition of Congruent Angles.
m/4 = m/9
5. m/2 = m/9 5. Transitive Property of Equality.
6. /2 > /9 6. Definition of Congruent Angles.
14. Statement Reason
1. DE || BC 1. Given
2. /2 > /AED 2. Postulate 11 - If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
3. /1 > /DEB 3. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
4. m/2 = m/AED 4. Definition of Congruent Angles.
m/1 = m/DEB
5. m/1 = m/2 5. Given
6. m/AED = m/DEB 6. Substitution of Equals.
7. /AED > /DEB 7. Definition of Congruent Angles.
8. DE is Between DA and EB 8. Definition of Betweeness (Rays - Page 131).
9. m/AED + m/DEB = m/AEB 9. Postulate 7 - Protractor - Fourth Assumption - Angle
Addition Assumption.
10. ED Bisects /AEB 10. Definition of Angle Bisector.
15. Statement Reason
1. AC Bisects /DAB 1. Given
2. /DAN > /CAB 2. Definition of Angle Bisector.
3. EN || AB 3. Given
4. /CAB > /ENA 4. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
5. m/DAN = m/CAB 5. Definition of Congruent Angles.
m/CAB = m/ENA
6. m/DAN = m/ENA 6. Transitive Property of Equality.
7. /DAN > /ENA 7. Definition of Congruent Angles.
Unit III — Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 285 – Lesson 3 — Theorem 17: If two parallel lines are cut by a transversal, then interior angles on the same side
of the transversal are supplementary
1. a) Theorem 17 - If two parallel lines are cut by a transversal, then interior angles on the same side
of the transversal are supplementary
b)
c) Given: Line , is parallel to line m cut by transversal t.
d) Prove: Interior Angles on the same side of the transversal are supplementary.
e) Statement Reason
1. , || m ; , and m are cut by transversal t. 1. Given
2. /6 > /2 2. Postulae 11 - If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
3. m/6 = m/2 3. Definition of Congruent Angles.
4. /2 and /4 are a linear pair 4. Definition of Linear Pair - exterior sides of angles
are opposite rays.
5. /2 and /4 are supplementary 5. Definition of Supplementary Angles. (Linear Pair).
6. m/2 + m/4 = 180 6. Definition of Supplementary Angles (Sum is 180).
7. m/6 + m/4 = 180 7. Substitution (3 into 6).
8. /6 and /4 are supplementary 8. Definition of Supplemenary Angles.
2. Statement Reason
1. , || m; , and mare cut by transversal t. 1. Given
2. /5 > /1 2. Postulate 11 - If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
3. m/5 = m/1 3. Definition of Congruent Angles.
4. /3 and /1 are a linear pair 4. Definition of Linear Pair - Exterior sides of angles
are opposite rays.
5. /3 and /1 form a straight angle 5. Definition of straight Angles - an angle whose sides
are opposite rays.
6. m/3 + m/1 = 180 6. Definition of Straight Angle - Giving a measure of 180.
7. m/3 + m/5 = 180 7. Substitution (3 into 7).
8. /3 and /5 are supplementary angles 8. Definition of Supplemenary Angles.
3. Theorem 17 - Corollary - If two parallel lines are cut by a transversal, then exterior angles on the same side
of the transversal are supplementary
1 2
3 4
5 6
7 8
<
m
t
65Part H – Theorems About Parallel Lines
66 Unit III – Fundamental Theorems
3. Continued
Given: Line , is parallel to line m cut by transversal t.
Prove: Exterior Angles on the same side of the transversal are supplementary.
Statement Reason
1. , || m ; , and m are cut by transversal t. 1. Given
2. Angles 4 and 6 are Supplementary Angles 2. Theorem 17 - If two parallel lines are cut by a transversal,
then interior angles on the same side of the transversal
are supplementary.
3. m/4 + m/6 = 180 3. Definition of Supplementary Angles.
4. /1 > /4 ; /7 > /6 4. Theorem 15 - If two lines intersect, then the vertical
angles formed are congruent.
5. m/1 = m/4 ; m/7 = m/6 5. Definition of Congruent Angles.
6. m/1 + m/7 = 180 6. Substitution of Equals (5 into 3).
7. /1 and /7 are Supplementary Angles 7. Definition of Supplementary Angles.
4. Statement Reason
1. , || m; , and m are cut by transversal t. 1. Given
2. Angles 3 and 5 are Supplementary Angles 2. Theorem 17 - If two parallel lines are cut by a transversal,
then interior angles on the same side of the transversal
are supplementary.
3. m/3 + m/5 = 180 3. Definition of Supplementary Angles.
4. /2 > /3 ; /8 > /5 4. Theorem 15 - If two lines intersect, then the vertical
angles formed are congruent.
5. m/2 = m/3 ; m/8 = m/5 5. Definition of Congruent Angles.
6. m/2 + m/8 = 180 6. Substitution of Equals (5 into 3).
7. /2 and /8 are Supplementary Angles 7. Definition of Supplementary Angles.
5.
/ABC and /DCB are Interior Angles on the same side of Transversal BC of two parallel lines. /ABC is a right angle.
m/ABC + m/DCB = 180
90 + 3x = 180
90 + 3x - 90 = 180 - 90
3x = 90
1/3 • 3 x = 1/3 • 90
x = 30
/BAD and /CDA are Interior Angles on the same side of Transversal AD of two parallel lines
1 2
3 4
5 6
7 8
<
m
t
5. Continued
m/BAD + m/CDA = 180
y - 10 + 100 = 180
y + 90 = 180
y + 90 - 90 = 180 - 90
y = 90
6.
/MNP and /NPQ are Interior Angles on the same side of Transversal NP of two parallel lines.
m/MNP + m/NPQ = 180
m/MNP + 132 = 180
m/MNP + 132 - 132 = 180 - 132
m/MNP = 48
m/1 = 1/2 m/MNP 1/2 • 48 = 24
m/2 = 1/2 m/MNP 1/2 • 48 = 24
/NRP > /4 - Alternate Interior Angles
m/4 = 1/2 • m/NPQ = 1/2 • 132 = 66. Therefore, m/NRP = 66
7. m/5 = 100 Theorem 17 - Interior Angles Supplementary
8. m/5 = 73 Theorem 16 - Alternate Interior Angles Congruent
9. m/5 = 104 Postulate 11 - Corresponding Angles Congruent
10. m/5 = 76 Theorem 15 - Vertical Angles Congruent
11. m/5 = 82 /5 > /4 - Alternate Interior Angles Congruent
/4 is supplementary to /3 - Linear Pair
12. m/5 = 120 Theorem 17 - Interior Angles Supplementary
m/5 + m/2 = 180
2 • m/2 + m/2 = 180
3 • m/2 = 180
m/2 = 60
m/5 = 2 • m/2 = 2 • 60 = 120
13. Statement Reason
1. m || n 1. Given
2. /ABG is Supplementary to /HED 2. Theorem 17 - If two parallel lines are cut by a transversal,
then interior angles on the same side of the transversal
are supplementary.
3. m/ABG + m/HED = 180 3. Definition of Supplementary Angles.
4. /HED > /FEG 4. Theorem 15 - If two lines intersect, then the vertical
angles formed are congruent.
5. m/HED = m/FEG 5. Definition of Congruent Angles.
6. m/ABG + m/FEG = 180 6. Substitution of Equals (5 into 3).
7. /ABG is supplementary to /FEG 7. Definition of Supplementary Angles.
67Part H – Theorems About Parallel Lines
68 Unit III – Fundamental Theorems
14. Statement Reason
1. AB || CD 1. Given
2. /BAC and /DCA are Supplementary Angles 2. Theorem 17 - If two parallel lines are cut by a transversal,
then interior angles on the same side of the transversal
are supplementary.
3. m/BAC + m/DCA = 180 3. Definition of Supplementary Angles.
4. m/BAC = m/BAE + m/1 4. Postulate 7 - Protractor - Fourth Assumption -
m/DCA = m/DCE + m/2 “Angle Addition” Postulate.
5. m/BAE + m/1 + m/DCE + m/2 = 180 5. Substitution (5 into 3).
6. m/1 and m/2 are Complementary Angles 6. Given
7. m/1 + m/2 = 90 7. Definition of Complementary Angles.
8. m/BAE + m/1 + m/DCE + m/2 8. Subtraction Property of Equality ( 5-7).
- ( m/1 + m/2) = 180 - 90
9. m/BAE + m/1 + m/DCE + m/2 + 9. Definition of Subtraction (a - b means a + - b)
- (m/1 + m/2) = 180 - 90
10. m/BAE + m/1 + m/DCE + m/2 + 10. Definition of Opposite (- (a) means - 1 • a)
- 1 3 (m/1 + m/2) = 180 - 90
11. m/BAE + m/1 + m/DCE + m/2 + 11. Distributive Property of Multiplication- 1 3 m/1 - 1x m/2 = 180 - 90 Over Additon.
12. m/BAE + m/1 + m/DCE + m/1 + 12. Property of (- 1) for Multiplication (-1 • a = - a).
- m/1 - m/2 = 180 - 90
13. m/BAE + m/DCE + m/1 + - m/1 + 13. Commutative Property of Addition (twice).
+ m/2 + - m/2) = 180 - 90
14. m/BAE + m/DCE + 0 + 0 = 180 - 90 14. Additive Inverse Property ( a + - a = 0) - (twice).
15. m/BAE + m/DCE = 180 - 90 15. Identity Property for Addition (a + 0 = a) - (twice).
16. m/BAE + m/DCE = 90 16. Substitution ( 180 - 90 = 90 )
17. m/BAE and /DCE are Complementary Angles 17. Definition of Complementary Angles.
18. AE bisects /BAC 18. Given
19. /BAE > /1 19. Definition of Angle Bisector.
20. m/BAE = m/1 20. Definition of Congruent Angles.
21. m/BAE + m/2 = 90 21. Substitution ( 20 into 7).
22. /BAE and /2 are Complementary Angles 22. Definition of Complementary Angles.
23. /DCE > /2 23. Theorem 13 - If two angles are complementary to the
same angle, then they are congruent to each other.
24. CE is Between CA and CD 24. Definition of Betweeness for Rays ( or segments) - CA,
CD, and CE are co-planar, and A < E < D or D < E < A
on the three segments.
25. CE bisects /DCA 25. Definition of Angle Bisector - A ray (segment) which is
between the sides of an angle, and divides the angle
into two congruent angles.
15. Statement Reason
1. EF || CB 1. Given
2. /EFD and /FDC are Supplementary Angles. 2. Theorem 17 - If two parallel lines are cut by a transversal,
then interior angles on the same side of the transversal
are supplementary.
3. m/EFD + m/FDC = 180 3. Definition of Supplementary Angles.
4. FD || AC 4. Given
5. /AEF is congruent to /EFD 5. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
6. m/AEF = m/EFD 6. Definiton of Congruent Angles.
7. m/AEF + m/FDC = 180 7. Substitution ( 6 into 3 )
8. /AEF is Supplementary to /FDC 8. Definition of Supplementary Angles.
Unit III — Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 287 – Lesson 4 — Theorem 18: If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
1. a) Theorem 18 - If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
b)
c) Given: Line , is parallel to line m.
Transversal t is perpendicular to line ,d) Prove: Transversal t is perpendicular to line m.
e) Statement Reason
1. , || m ; t ,1. Given
2. /1 is a right angle. 2. Definition of Perpendicular Lines
3. m/1 = 90 3. Definition of a Right Angle.
4. /1 > /5 4. Postulate 11 - If two parallel lines are cut by a transversal,
then corresponding angles are congruent.
5. m/1 = m/5 5. Definition of Congruent Angles.
6. 90 = m/5 6. Substitution of Equals - ( 3 into 5 )
7. /5 is a right angle 7. Definition of Right Angle.
8. Transversal t is perpendicular to line m 8. Definition of Perpendicular Lines.
2. m/AEK = 87º - contradicts the other information. All angles in this diagram must be right angles.
3. m/DAE = 43 , m/EAB = 47 contradicts AE Bisects /DAB . AE Bisects /DAB means /DAE /EAB.
4. No contradictory information
5. /JHM and /DLM cannot be supplementary since m/ JHM must be 90 and we do not know what m/DLM is. AE || GM,
CJ AF means CJ is also perpendicular to GM.
1 2
3 4
5 6
7 8
<
m
t
69Part H – Theorems About Parallel Lines
70 Unit III – Fundamental Theorems
6. a) 90 ; Theorem 18 - If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
b) 30 , Theorem 16 - If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
c) 60 ; /DBA , /ABC , and /CBE form a straight angle
90 + 30 + m/CBE = 180
m/CBE = 60
d) 60 ; Postulate 11 ; If two parallel lines are cut by a transversal, then corresponding angles are congruent.
e) 60 ; /CAB and /BAD are complementary.
7. Statement Reason
1. AB || DC ; DA AB 1. Given
2. DA DC 2. Theorem 18 - If a given line is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
3. /DAB is a right angle 3. Definition of Perpendicular.
/ADC is a right angle
4. /DAB > /ADC 4. Theorem 11 - If you have right angles, then those right
angles are congruent.
5. m/DAB = m/ADC 5. Definition of Congruent Angles.
6. /AXZ > /DZX 6. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
7. m/AXZ = m/DZX 7. Definition of Congruent Angles.
8. m/DYX = m/DAB + m/AXZ 8. Given
9. m/DYX = m/ADC + m/DZX 9. Substitution ( 5 and 7 into 8 )
8. Statement Reason
1. QR || MN ; PM MN 1. Given
2. PM QR 2. Theorem 18 - If a given line is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
3. m/PQR is a right angle 3. Definition of Perpendicular.
m/PMN is a right angle
4. m/PQR = 90 4. Definition of a Right Angle.
m/PMN = 90
5. m/PQR + m/PMN = 90 + 90 5. Addition Property for Equality.
6. m/PQR + m/PMN = 180 6. Substitution of Equals ( 90 + 90 = 180 ).
7. /PQR and /PMN are supplementary 7. Definition of Supplementary Angles
9. Statement Reason
1. PC || QD 1. Given
AB QD
2. AB PC 2. Theorem 18 - If a given line is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
3. /CPB is a right angle 3. Definition of Perpendicular.
4. m/CPB = 90 4. Definition of a Right Angle.
9. continued Statement Reason
5. PR bisects /CPB 5. Given
6. /2 > /1 6. Definition of Angle Bisector
7. m/2 = m/1 7. Definition of Congruent Angles.
8. m/1 + m/2 = m/CPB 8. Postulate 7 - Protractor - Fourth Assumption -
“Angle Additon” Assumption.
9. m/1 + m/1 = m/CPB 9. Subsitution ( 7 into 8)
10. ( 1 + 1 ) • m/1 = m/CPB 10. Distributive Property of Multiplication over Addition.
11. 2 • m/1 = m/CPB 11. Substitution ( 1 + 1 = 2).
12. 2m/1 = 90 12. Substitution ( 4 into 11 ).
13. 1/2 • 2 m/1 = 1/2 • 90 13. Multiplication Property for Equality.
14. 1 • m/1 = 1/2 • 90 14. Multiplicative Inverse Property.
15. m/1 = 1/2 • 90 15. Identity Property for Multiplication.
16. m/1 = 45 16. Substitution ( 1/2 • 90 = 45 )
10. a) 72 f) 18
b) 90 g) 72
c) 72 h) 108
d) 18 i) 72
e) 90 j) 108
11. Statement Reason
1. RU || SV, RS || UT 1. Given
RS RU
2. RS SW 2. Theorem 18 - If a given line is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
3. /RSW is a right angle 3. Definition of Perpendicular Lines.
4. m/RSW = 90 4. Definition of Right Angle.
5. m/RSU + m/USW = m/RSW 5. Postulate 7 - Protractor - Fourth Assumption -
“Angle Additon” Assumption.
6. m/USW = 55 6. Given
7. m/RSU + 55 = m/RSW 7. Subsitution ( 6 into 5 ).
8. m/RSU + 55 = 90 8. Subsitution ( 4 into 7 ).
9. m/RSU + 55 + - 55 = 90 + - 55 9. Additon Property for Equality.
10. m/RSU + 0 = 90 + - 55 10. Additive Inverse Property.
11. m/RSU = 90 + - 55 11. Identity Property of Addition
12. m/RSU = 35 12. Substitution ( 90 + - 55 = 35 ))
13. /RSU > /SUT 13. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
14. m/RSU = m/SUT 14. Definition of Congruent Angles.
15. 35 = m/SUT 15. Substitution ( 12 into 14 ).
71Part H – Theorems About Parallel Lines
72 Unit III – Fundamental Theorems
11. continued Statement Reason
16. TU bisects /SUV 16. Given
17. /SUT > /TUV 17. Definition of Angle Bisector.
18. m/SUT = m/TUV 18. Definition of Congruent Angles.
19. 35 = m/TUV 19. Substitution ( 15 into 18 ).
20. m/TUV = 35 20. Symmetric Property of Equality
Statement Reason
12.
1. AC || UW; AD || UX 1. Given
EY AC
2. EY UW 2. Theorem 18 - If a given line is perpendicular to one of
two parallel lines, then it is perpendicular to the other.
3. /ABE is a right angle 3. Definition of Perpendicular Lines.
/UVY is a right angle
4. /ABE > /UVY 4. Theorem 11 - If you have right angles, then those right
angles are congruent.
5. m/ABE = m/UVY 5. Definition of Right Angle.
6. /YDA > /EXU 6. Theorem 16 - If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
7. m/YDA = m/EXU 7. Definition of Congruent Angles.
8. m/YDA + m/ABE = m/EXU + m/UVY 8. Addition Property of Equality ( 7 + 5 ).
73Part H – Theorems About Parallel Lines
Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 291 – Lesson 5 — Theorem 19: If two lines are cut by a transversal so that corresponding angles are congruent,
then the two lines are parallel.
1. a) Theorem 19: If two lines are cut by a transversal so that corresponding angles are congruent, then the two lines are parallel.
b)
c) Given: Line < and line m are cut by transversal t
/1 > /5
d) Prove: < || m
e) Statement Reason
1. Line < and line m are cut by transversal t 1. Given
/1 >/5
2. m/1 = m/5 2. Definition of Congruent Angles
3. Suppose < ||y m 3. Indirect Proof Assumption
4. Draw auxiliary line <1 through the intersection 4. Postulate 9 – In a plane, through a point
point of line < and transversal t so that <1 || m. not on a given line, there is exactly one line
parallel to the given line.
5. /11 > /5 5. Postulate 11 – If two parallel lines are cut
by a transversal, then corresponding angles
are congruent.
6. m/11 = m/5 6. Definition of Congruent Angles
7. m/11 = m/1 7. Substitution (2 into 6)
8. But m/11 ≠ m/1 8. Postulate 7 – Protractor Second Assumption –
To every pair of rays with a common endpoint,
there corresponds exactly one real number from
0 to 180, inclusive, called the unique measure
of the angle formed by the rays.
9. Our assumption is false, and 9. Reductio ad Absurdum
line < is parallel to line m.
2. a and c 6. a and d 10. no 14. yes
3. b and d 7. b and c 11. yes 15. yes
4. a and c 8. a and d 12. yes 16. no
5. b and d 9. a and c 13. yes 17. yes
1 2
3 4
5 6
7 8
<
m
t
<1
18. Statement Reason
1. BC##$ || EF##$ 1. Given
2. /3 > /2 2. Postulate 11 – If two parallel lines are cut
by a transversal, then corresponding angles
are congruent.
3. /1 > /3 3. Given
4. m /1 = m /3 4. Definition of Congruent Angles
m /3 = m /2
5. m /1 = m /2 5. Transitive Property of Equality
6. /1 > /2 6. Definition of Congruent Angles
7. BA##$ || ED##$ 7. Theorem 19 – If two lines are cut by a transversal
so that corresponding angles are congruent, then
the two lines are parallel.
19. Statement Reason
1. BC### || AE##$ 1. Given
2. /1 > /3 2. Theorem 16 – If two parallel lines are cut by
a transversal, then alternate interior angles
are congruent.
3. m/1 = m/3 3. Definition of Congruent Angles
4. /3 > /2 4. Given
5. m/3 = m/2 5. Definition of Congruent Angles
6. m/1 = m/2 6. Transitive Property of Equality
7. /1 > /2 7. Definition of Congruent Angles
8. DC### || AB### 8. Theorem 19 – If two lines are cut by a transversal
so that corresponding angles are congruent, then
the two lines are parallel.
20. Statement Reason
1. AB || CD 1. Given
2. /BEG > /DFG 2. Postulate 11 – If two parallel lines are cut
by a transversal, then corresponding angles
are congruent.
3. EH##$ bisects /BEG 3. Given
FJ##$ bisects /DFG
4. /BEH > /HEG 4. Definition of Angle Bisector
/DFJ > /JFG
5. m/BEG = m/DFG 5. Definition of Congruent Angles
m/BEH = m/HEG
m/DFJ = m/JFG
6. m/BEG = m/BEH + m/HEG 6. Postulate 7 – Protractor Fourth Assumption –
m/DFG = m/DFJ + m/JFG
7. m/BEH + m/HEG = m/DFJ + m/JFG 7. Substitution of Equals (6 into 5)
74 Unit III – Fundamental Theorems
Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 294 – Lesson 6 — Theorem 20: If two lines are cut by a transversal so that alternate interior angles are congruent,
then the two lines are parallel.
1. a) Theorem 20: If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.
b)
c) Given: Line < and line m are cut by transversal t
/3 > /6
d) Prove: < || m
e) Statement Reason
1. Line < and line m are cut by transversal t 1. Given
2. /3 > /6 2. Given
3. m/3 = m/6 3. Definition of Congruent Angles
4. /2 and /3 are vertical angles 4. Definition of Vertical Angles – Two angles
with a common vertex whose sides are
opposite rays.
5. /2 > /3 5. Theorem 15 – If two lines intersect, then
the vertical angles formed are congruent.
6. m/2 = m/3 6. Definition of Congruent Angles
7. m/2 = m/6 7. Substitution of Equals (6 into 3)
8. /2 > /6 8. Definition of Congruent Angles
9. < || m 9. Theorem 19 – If two lines are cut by a transversal
so that corresponding angles are congruent, then
the two lines are parallel.
2. a) Corollary 20a: If two lines are cut by a transversal so that alternate external angles are congruent, then the two lines are parallel.
b)
1 2
3 4
5 6
7 8
<
m
t
1 2
3 4
5 6
7 8
<
m
t
76 Unit III – Fundamental Theorems
77Part H – Theorems About Parallel Lines
2. (continued)
c) Given: Line < and line m are cut by transversal t
/3 > /6
d) Prove: < || m
e) Statement Reason
1. Line < and line m are cut by transversal t 1. Given
2. /1 > /8 2. Given
3. m/1 = m/8 3. Definition of Congruent Angles
4. /1 and /4 are vertical angles 4. Definition of Vertical Angles – Two angles
/5 and /8 are vertical angles with a common vertex whose sides are
opposite rays.
5. /1 > /4 5. Theorem 15 – If two lines intersect, then
/5 > /8 the vertical angles formed are congruent.
6. m/1 = m/4 6. Definition of Congruent Angles
m/5 = m/8
7. m/4 = m/5 7. Substitution (6 into 3; 3 into 6)
8. /4 > /5 8. Definition of Congruent Angles
9. < ||m 9. Theorem 20 – If two lines are cut by a transversal
so that alternate interior angles are congruent,
then the two lines are parallel.
3. b and c 7. c and d 11. yes 15. yes
4. b and d 8. a and c 12. no 16. yes
5. a and d 9. b and d 13. no 17. yes
6. a and b 10. d and a 14. yes 18. yes
19. Statement Reason
1. /1 > /2 1. Given
2. BC### || FE### 2. Theorem 20 – If two lines (BC### and FE### ) are cut by
a transversal (AD### ) so that alternate interior angles
are congruent, then the two lines are parallel.
20. Statement Reason
1. AF##$ > AD### ; CD##$ > AD### 1. Given
2. /DAF is a right angle 2. Definition of Perpendicular Lines
/ADC is a right angle
3. /DAF > /ADC 3. Theorem 11 – If you have right angles,
then those right angles are congruent
4. AF##$ || CD##$ 4. Theorem 20 – If two lines (AF##$ and DC##$ ) are cut by
a transversal (AD### ) so that alternate interior angles
are congruent, then the two lines are parallel.
79Part H – Theorems About Parallel Lines
Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 297 – Lesson 7 — Theorem 21: If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, then the two lines are parallel.
1. a) Theorem 21: If two lines are cut by a transversal so that interior angles on the same side of the transversal
are supplementary, then the two lines are parallel.
b)
c) Given: Line < and line m are cut by transversal t
/3 is supplementary to /5
d) Prove: < || m
e) Statement Reason
1. Line < and line m are cut by transversal t 1. Given
2. /3 is supplementary to /5 2. Given
3. /1 and /3 form a linear pair 3. Definition of Linear Pair – Two angles which have
a common side (they are adjacent), and whose
exterior sides are opposite rays.
4. /1 is supplementary to /3 4. Theorem 10 – If the exterior sides of two adjacent
angles are opposite rays, then the angles are
supplementary.
5. /1 > /5 5. Theorem 14 – If two angles are supplementary
to the same angle, then they are congruent
to each other.
6. < || m 9. Theorem 19 – If two lines are cut by a transversal
so that corresponding angles are congruent, then
the two lines are parallel.
2. a) Corollary 21a: If two lines are cut by a transversal so that exterior angles on the same side of the transversal
are supplementary, then the two lines are parallel.
b)
1 2
3 4
5 6
7 8
<
m
t
1 2
3 4
5 6
7 8
<
m
t
2. (continued)
c) Given: Line < and line m are cut by transversal t
/2 is supplementary to /8
d) Prove: < || m
e) Statement Reason
1. Line < and line m are cut by transversal t 1. Given
2. /2 is supplementary to /8 2. Given
3. m/2 + m/8 = 180 3. Definition of Supplementary Angles
4. /2 > /3 4. Theorem 15 – If two lines intersect, then
/8 > /5 the vertical angles formed are congruent.
5. m/2 = m/3 5. Definition of Congruent Angles
m/8 = m/5
6. m/3 + m/5 = 180 6. Substitution (5 into 3)
7. /3 is supplementary to /5 7. Definition of Supplementary Angles
8. < || m 8. Theorem 21 – If two lines are cut by a transversal
so that interior angles on the same side of the
transversal are supplementary, then the two lines
are parallel.
3. a and c 5. b and d 7. c and d 9. b and c
4. b and d 6. c and d 8. a and d 10. a and d
11. 2x + 4x = 180
6x = 1801/6 • 6x = 1/6 • 180
x = 30
12. 2x + y + 120 = 180
+2x – y + 140 = 180
4x +0 •y + 260 = 360
4x + 260 – 260 = 360 – 260
4x = 1001/4 • 4x = 1/4 • 100
x = 25
2(25) + y + 120 = 180
y + 170 = 180
y + 170 – 170 = 180 – 170
y = 10
13. If AB### is to remain parallel to DC### , then
m/2 must decrease by 30 degrees.
14. m /1 +m /2 = 180
x2 + 8x + 4x + 20 = 180
x2 + 12x + 20 = 180
x2 + 12x + 20 – 180 = 180 – 180
x2 + 12x – 160 = 0
(x + 20) (x – 8) = 0
x + 20 = 0 or x – 8 = 0
x = –20 or x = 8
If x = 8, then m /2 = 4 (8) + 20 = 52, and
m /1 = (8)2 + 8 (8) = 64 + 64 = 128. Since
128 + 52 =180, x = 8 is an acceptable answer.
15. yes
16. no
17. yes
18. yes
80 Unit III – Fundamental Theorems
81Part H – Theorems About Parallel Lines
19. Statement Reason
1. /R and /S are supplementary angles 1. Given
2. m/R + m/S = 180 2. Definition of Supplementary Angles
3. /Q > /S 3. Given
4. m/Q = m/S 4. Definition of Congruent Angles
5. m/R + m/Q = 180 5. Substitution of Equals (4 into 2)
6. /R and /Q are supplementary angles 6. Definition of Supplementary Angles
7. QP### || RS### 7. Theorem 21 – If two lines are cut by a transversal
so that interior angles on the same side of the
transversal are supplementary, then the two lines
are parallel.
20. Statement Reason
1. /BCF > /BFE 1. Given
2. /BFE > /BFC 2. Given
3. FC##$ bisects /BFG 3. Given
4. /BFC > /CFG 4. Definition of Angle Bisector
5. m/BCF = m/BFE 5. Definition of Congruent Angles
m/BFE = m/BFC
m/BFC = m/CFG
6. m/EFC + m/CFG = m/EFG 6. Postulate 7 – Protractor - Fourth Assumption –
7. /EFG is a straight angle 7. Definition of Straight Angle –
An angle whose sides are opposite rays …
8. m/EFG = 180 8. Definition of Straight Angle –
…giving a measure of 180 degrees.
9. m/EFC + m/CFG = 180 9. Substitution of Equals (6 into 8)
10. m/BCF = m/CFG 10. Transitive Property of Equality
11. m/EFC + m/BCF = 180 11. Substitution of Equals (10 into 9)
12. /EFC and /BCF are supplementary angles 12. Definition of Supplementary Angles
13. AD@#$ || EG@#$ 13. Theorem 21 – If two lines are cut by a transversal
so that interior angles on the same side of the
transversal are supplementary, then the two lines
are parallel.
21. Statement Reason
1. /ABC > /CBD; /ABC > /EFH 1. Given
2. m/ABC = m/CBD 2. Definition of Congruent Angles
m/ABC = m/EFH
3. m/EFH = m/CBD 3. Substitution of Equals (2 into 2)
4. /EFH > /CBD 4. Definition of Congruent Angles
5. AD@#$ || EG@#$ 5. Corollary 20a – If two lines are cut by a transversal
so that alternate exterior angles are congruent,
then the two lines are parallel.
22. Statement Reason
1. /BCD > /D 1. Given
2. m/BCD = m/D 2. Definition of Congruent Angles
3. m/B + m/D = 180 3. Given
4. m/B + m/BCD = 180 4. Substitution of Equals (2 into 3)
5. /B and /BCD are supplementary angles 5. Definition of Supplementary Angles
6. BA##$ || CD### 6. Theorem 21 – If two lines are cut by a transversal
so that interior angles on the same side of the
transversal are supplementary, then the two lines
are parallel.
23. Statement Reason
1. Either < ||y m. or < || m.. 1. Indirect Proof Assumption
Assume < || m..
2. /2 and /4 are supplementary angles 2. Theorem 17 – If two parallel lines are cut by
a transversal, then interior angles on the same
side of the transversal are supplementary.
3. /2 and /4 are not supplementary 3. Given
4. < ||y m 4. R.A.A. – Statements 2 and 3 contradict each other,
so our indirect proof assumption is false.
82 Unit III – Fundamental Theorems
83Part H – Theorems About Parallel Lines
Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 301 – Lesson 8 — Theorem 22: If two lines are perpendicular to a third line, then the two lines are parallel.
1. a) If two lines are perpendicular to a third line, then the two lines are parallel.
b)
c) Given: < > t
m > t
d) Prove: < || m
e) Statement Reason
1. < > t ; m > t 1. Given
2. /2 is a right angle 2. Definition of a Right Angle
/6 is a right angle
3. /2 > /6 3. Theorem 11 – If you have right angles, then
those right angels are congruent.
4. < || m 4. Theorem 19 – If two lines are cut by a transversal
so that corresponding angles are congruent, then
the two lines are parallel.
2. Statement Reason
1. t > <1 1. Given
2. <1 || <3 2. Given
3. t > <3 3. Theorem 18 – If a given line is perpendicular to
one of two parallel lines, then it is perpendicular
to the other.
4. <2 || <3 4. Given
5. t > <2 5. Theorem 18 – If a given line is perpendicular to
one of two parallel lines, then it is perpendicular
to the other.
3. Statement Reason
1. DE### > BC### 1. Given
2. m /B = 90 2. Given
1 2
3 4
5 6
7 8
<
m
t
3. (continued)
Statement Reason
3. /B is a right angle 3. Definition of Right Angle
4. AB### > BC### 4. Definition of Perpendicular Lines
5. AB### || DE### 5. Theorem 22 – Two lines are perpendicular to
a third line, then the two lines are parallel.
4. Statement Reason
1. /H and /HEF are supplementary angles 1. Given
2. HG### || FE### 2. Theorem 21 – If two lines are cut by a transversal
so that interior angles on the same side of the
transversal are supplementary, then the two lines
are parallel.
3. EG### > HG### 3. Given
4. EG### > FE### 4. Theorem 18 – If a given line is perpendicular to
one of two parallel lines, then it is perpendicular
to the other.
5. Statement Reason
1. /BDE is a right angle 1. Given
2. BD### > DE### 2. Definition of Perpendicular Lines
3. AB### > BD### 3. Given
4. AB### || DE### 4. Theorem 22 – If two lines are perpendicular to
a third line, then the two lines are parallel.
5. /A > /E 5. Theorem 16 – If two parallel lines are cut by
a transversal, then alternate interior angles
are congruent.
6. m/A = m/E 6. Definition of Congruent Angles
6. Statement Reason
1. /AEB and /CFD are right angles t1. Given
2. AG### > CB### 2. Definition of Perpendicular Lines
HD### > CB###
3. AG### || HD### 3. Theorem 22 – If two lines are perpendicular to
a third line, then the two lines are parallel.
4. /ACG > /HDC 4. Postulate 11 – If two parallel lines are cut
by a transversal, then corresponding angles
are congruent.
84 Unit III – Fundamental Theorems
85Part H – Theorems About Parallel Lines
7. Statement Reason
1. t > < ; < is not parallel to m 1. Given
2. Either t > m or t >y m. 2. Indirect Proof Assumption
Assume t > m.
3. < || m 3. Theorem 22 – If two lines are perpendicular to
a third line, then the two lines are parallel.
4. This contradicts the given < ||y m. So 4. R.A.A.
t > m is false, and t >y m must be true.
8. m/A = 47
a) AB || CD Theorem 22 – If two lines are perpendicular to a third line,
then the two lines are parallel.
b) /A > /D Theorem 16 – If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
9. AB || CD Theorem 22 – If two lines are perpendicular to a third line,
then the two lines are parallel.
m/ABE + m/BEC = 180 Theorem 17 – If two parallel lines are cut by a transversal
37 + m/BEC = 180 then interior angles on the same side of the transversal are supplementary.
m/BEC = 143
/ABE > /BED Theorem 16 – If two parallel lines are cut by a transversal,
37 = m/BED then alternate interior angles are congruent.
10. m/BAC = 55 Linear Pair of angles is supplementary.
m/DCE = 125Theorem 22 – If two lines are perpendicular to a third line,
then the two lines are parallel.
Theorem 16 Corollary – If two parallel lines are cut by a transversal,
then alternate exterior angles are congruent.
11. 4 If the line intersects the plane, then 3 points determine the plane,
plus 1 point outside the plane can be paired with one of the three points
to determine the line.
12. AB@#$ The planes intersect with the line AB being the intersection.
Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 304 – Lesson 9 — Theorem 23: If two lines are parallel to a third line, then the two lines are parallel to each other.
1. a) Theorem 23 – If two lines are parallel to a third line, then the two lines are parallel to each other.
b)
c) Given: < || r ; m || r
d) Prove: < || m
e) Statement Reason
1. < || r ; m || r 1. Given
2. Assume < ||ym 2. Indirect Proof Assumption
3. Line < must intersect line m at a point P. 3. If two coplanar lines are not parallel, then they
do intersect. (Contrapositive of Definition of
Parallel Lines)
4. Since line m passes through point P (and is 4. This is not possible by Postulate 9.Through a
parallel to line r ) and line < passes through point not on a line, there can be only one line
point P (and is parallel to line r ), we have two parallel to a given line.
lines through the same point parallel to line r .
5. < || m 5. R.A.A.
2. a || d 4. e || c 6. a || b
3. e || d 5. d || c 7. a || e
8. <1 || <3 Theorem 20 – Alternate Interior Angles Congruent
<1 || <2 Theorem 21 – Same Side Interior Angles Supplementary
<2 || <3 Theorem 23 – Two lines parallel to a third, or
Theorem 15 – Vertical Angles Congruent; Theorem 21 – Same Side Interior Angles Supplementary
t 2 || t 4 Theorem 20 Corollary – Alternate Exterior Angles Congruent
9. Yes c || b Theorem 21 – Same Side Interior Angles Supplementary
c || a Theorem 19 – Corresponding Angles Congruent
Therefore, a || b . Theorem 23
<
mP
r
•
86 Unit III – Fundamental Theorems
87Part H – Theorems About Parallel Lines
10. no b || c Theorem 22 – Two lines perpendicular to third line are parallel to each other.
b || a Theorem 20 Corollary – Alternate Exterior Angles Congruent
Therefore, a || c. Theorem 23
11. no b || c Theorem 19 – Corresponding Angles Congruent
b || a Theorem 21 Corollary – Same Side Exterior Angles Supplementary
Therefore, a || c. Theorem 23
12. yes a || c Theorem 21 Corollary – Same Side Exterior Angles Supplementary
b || c Theorem 20 Corollary – Alternate Exterior Angles Congruent
Therefore, a || b. Theorem 23
13. Statement Reason
1. /1 > /2 1. Given
2. p || q 2. Theorem 20 – If two lines are cut by a transversal
so that alternate interior angles are congruent,
then the two lines are parallel.
3. q || r 3. Given
4. p || r 4. Theorem 23 – If two lines are parallel to a third
line, then the two lines are parallel to each other.
14. Statement Reason
1. Either line t intersects line q or line t does 1. Indirect Proof Assumption
not intersect line q. Assume line t does not
intersect line q at point R .
2. t || q and t contains point R . 2. Definition of Parallel Lines – Lines which are
coplanar and do not intersect
3. p ||q ; transversal t intersects line p at point R . 3. Given
4. t || p is a contradition 4. Theorem 23 – If two lines are parallel to a third
line, then the two lines are parallel to each other.
5. The assumption that t does not intersect 5. R.A.A.
q is false. we conclude t intersects q.
15. Statement Reason
1. Draw auxiliary line ZW@##$ parallel to UV@#$ where 1. Postulate 9 – In a plane, through a point not on a
point Z is on the U side of XW@##$ a given line, there is exactly one line parallel to
the given line.
2. UV@#$ || XY@#$ 2. Given
15. (continued)
Statement Reason
3. ZW@##$ || XY@#$ 3. Theorem 23 – If two lines are parallel to a third
line, then the two lines are parallel to each other.
4. /VUW > /ZWU 4. Theorem 16 – If two parallel lines are cut by
/WXY > /XWZ a transversal, then alternate interior angles
are congruent.
5. m/VUW = m/ZWU 5. Definition of Congruent Angles
m/WXY = m/XWZ
6. m/XWU = m/XWZ + m/ZWU 6. Postulate 7 – Protractor - Fourth Assumption –
“Angle Addition” Assumption
7. m/XWU = m/WXY + m/VUW 7. Substitution of Equals (5 into 6)
8. m/XWU = m/VUW + m/WXY 8. Commutative Property of Addition
16. Theorem 23 would be true when the three lines are in space and when the three lines are in the same plane as assumed.
A simple example would be the lines determined by the gable and two edges of the roof of a small rectangular building.
Essentially, three planes are determined by the three non-coplanar lines where each pair of parallel lines determine one plane.
17. Theorem 22 Rewritten – If two planes are perpedicular to a third plane, then the two planes are parallel.
This statement is not always true. In an ordinary room, if the two planes are the floor and ceiling perpendicular to a wall, then
the statement is true. If however, the two planes are two intersecting walls perpendicular to the floor, then the statement is
not true.
Theorem 23 Rewritten – If two planes are parallel to a third plane, then the two planes are parallel to each other.
This statement is true. Think of the three planes being the floors of a three story building.
18. Statement Reason
1. /1 > /3 1. Given
2. p || q 2. Theorem 19 – If two lines are cut by a transversal
so that corresponding angles are congruent, then
the two lines are parallel.
3. p || r 3. Given
4. q || r 4. Theorem 23 – If two lines are parallel to a third
line, then the two lines are parallel to each other.
5. /3 and /4 are supplementary angles 5. Theorem 17 – If two parallel lines are cut by a
transversal, then interior angles on the same side
of the transversal are supplementary.
88 Unit III – Fundamental Theorems
89Part H – Theorems About Parallel Lines
Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines
p. 307 – Lesson 10 — Theorem 24: If two parallel planes are cut by a third plane, then the two lines of intersection are parallel.
1. a) Theorem 24: If two parallel planes are cut by a third plane, then the two lines of intersection are parallel.
b)
c) Given: Plane P is parallel to plane Q.
Plane R intersects plane P and plane Q in lines m and n.
d) Prove: Line m is parallel to line n.
e) Statement Reason
1. Plane P is parallel to plane Q. 1. Given
2. Plane R intersects plane P and plane Q 2. Given
in lines m and n.
3. Lines m and n are straight lines. 3. Postulate 5 – If two planes intersect, then the
intersection is a unique line.
4. Lines m and n lie in the same plane R. 4. Step 2 – Given.
5. Line m (in plane P) and line n (in plane Q) 5. Step 1 – Plane P and plane Q are parallel, so
have no points in common. the planes have no points in common.
6. m || n 6. Definition of Parallel Lines – lines which are
coplanar and do not intersect.
2. a) Through a given point not on a given line, one and only one plane can be passed parallel to the given line.
False – not “one and only one”; infinitely many.
b) Through a given point not on a given plane, one and only one line can be drawn parallel to the given plane.
False – not “one and only one”; infinitely many.
c) If one of two perpendicular lines is parallel to a plane, the other is also parallel to the plane.
False – only one case; infinitely many others.
d) Two lines parallel to the same plane are perpendicular. False – only one case; Two lines can be in an infinite number
of positions in space, be parallel to each other, and at the same time be parallel to a given plane.
e) Two planes perpendicular to a third plane re perpendicular to each other.
False – only one case (box corner); infinitely many where the two given planes are not perpendicular.
f) Two planes parallel to the same line are perpendicular.
False – only one case; infinitely many cases where two planes are parallel.
Q
Rm
n
P
3. a) always true c) always true e) sometimes true g) always true
b) sometimes true d) always true f) sometimes true h) always true
4. a) /E – AB – L and /J – DC – L
/F – AB – L and /H – DC – L
b) /G – AB – E and /J – DC – K
/F – AB – G and /H – DC – K
c) /F – AB – G and /J – DC – L
/E – AB – G and /H – DC – L
also
/E – AB – L and /H – DC – K
/F – AB – L and /J – DC – K
5. 2) For any two different points, there is exactly one line containing them.
3) If two different planes intersect, the intersection is a unique line.
5) Definition of perpendicular lines
6) Definition of right angle
7) Addition property of equality
8) Substitution of equals ( 90 + 90 = 180)
9) Definition of supplementary angles
10) Theorem 21 – If two lines are cut by a transversal so that interior angles on the same side of the transversal are
supplementary, then the two lines are parallel.
6. 2) Postulate 5 – If two different planes intersect, the intersection is a unique line.
3) Definition of line perpendicular to a plane – If a line is perpendicular to a plane,
it is perpendicular to every line in the plane that intersects it.
7. 4) Definition of right angle
6) Definition of perpendicular planes – If two planes intersect and any line in one of them is perpendicular to their line of
intersection and to the other plane, then the two planes are perpendicular.
8. 3) Postulate 5 – If two different planes intersect, the intersection is a unique line.
7) Definition of perpendicular lines
8) Definition of right dihedral angle – the dihedral angle has plane angles that are right angles.
d) /G – AB – F and /E – AB – L
/E – AB – G and /F – AB – L
also
/H – DC – L and /J – DC – K
/H – DC – K and /J – DC – L
e) yes; yes
90 Unit III – Fundamental Theorems