Geometry – Unit 2 Targets & Info Name: This Unit’s theme – Reasoning and Proof September 9 – September 30 (Approximate Time for Test) Use this sheet as a guide throughout the chapter to see if you are getting the right information in reaching each target listed.

By the end of Unit 2, you should know how to… Target found in… Did I reach the target?

DIAGRAMS & EXAMPLES!

Identify and use correct vocabulary: Negation, inductive reasoning, deductive reasoning, converse, inverse, contrapositive, bisect, midpoint, perpendicular, complementary, supplementary, right angle

Chapter 2

Write a conditional statement in IF-THEN form along with its converse, inverse, and contrapositive, and determine if the statements are true or false.

Chapter 2 Section 2, pages 89-95

Write a biconditional statement as its conditional and converse statements and determine if the biconditional statement is true or false

Chapter 2 Section 3, pages 98-104

Justify statements with definitions, postulates, theorems proven in class, or properties

Chapter 2 Section 5, pages 113-119

Complete a two column proof by providing reasons that justify each given statement

Chapter 2 Sections 5 & 6

Use the Law of Detachment and Law of Syllogism to make valid conclusions

Chapter 2 Section 4 pages 106-112

Calculate the Surface Area and Volume of three-dimensional figures

All material covered on the test will be based on these targets. So keep track of your readiness for the test by updating the “Did I reach the target?” column.

Key Postulates, Properties, and Theorems: Segment Addition Postulate or Angle Addition Postulate Algebraic Properties (Addition, Subtraction, Multiplication, Division, Substitution, Distributive) Reflexive, Symmetric, Transitive Properties of Equality and Congruence Vertical angles are congruent. Linear pairs are supplementary. If two angles are congruent, then their supplements are congruent. If two angles are congruent, then their complements are congruent. All right angles are congruent.

Unit 2 Suggested Textbook Problems

Section 2.2 p. 93 #5 – 31 Section 2.3 p. 101 #7 – 38, 43 – 46 Section 2.4 p. 110 #6 – 17 Section 2.5 p. 117 #5 – 22 Section 2.6 p. 124 #6 – 20, 23

POSTULATES AND THEOREMS POSTULATES: Accepted without proof! Segment Addition Postulate If B is between A and C, then AB + BC = AC If AB + BC = AC, then B is between A and C. Angle Addition Postulate If P is in the interior of ∠RST, then m∠RSP + m∠PST = m∠RST. Through any two points there exists exactly one line. If two lines intersect, then their intersection is exactly one point. Through any three noncollinear points there exists exactly one plane. If two planes intersect, then their intersection is a line. If two angles form a linear pair, then they are supplementary.

THEOREMS: Proven true!

Theorem Picture/Rephrase

Lesson 1: Conditional Statements Conditional Statement: a logical statement beginning with a HYPOTHESIS and ending with a CONCLUSION. ***Normally, conditional statements are written in If-Then form.*** Example: If Jaiden wakes up at 2:30 am, then Nicole will be grouchy. HYPOTHESIS: Jaiden wakes up at 2:30 AM CONCLUSION: Nicole will be grouchy Underline the hypothesis and circle the conclusion in the following conditional statements. 1) If I win the lottery, then I will build a racket ball court and indoor tennis facility in Coal City. 2) If Mr. McCleary sees a Sandra Bullock movie, then he will be overexcited and not be able to sleep. 3) If two lines intersect, then their intersection is exactly one point. 4) There will be no more homework in Geometry when Mr. Leman can dunk a basketball unaided on a regulation hoop. Rewrite the following conditional statements in If-Then form. 1) Two points are collinear if they lie on the same line. 2) A number divisible by 9 is also divisible by 3. 3) Two planes intersect at a line. 4) Mr. Leman’s wife is the most beautiful woman in the world. Original Statement: If p, then q. Converse: If q, then p. Inverse: If not p, then not q. Contrapositive: If not q, then not p.

Example 1) Conditional: If m∠A = 30° , then m∠A is acute. Converse: If m∠A is acute, then m∠A = 30° . Inverse: Ifm∠A ≠ 30° , thenm∠A is not acute. Contrapositive: If m∠A is not acute, then m∠A ≠ 30° . Equivalent statements: statements that share the same truth value (T or F) ***A conditional statement and its contrapositive ALWAYS share the same truth value.*** ***The converse and inverse of a statement ALWAYS share the same truth value.*** Negation: writing the negative, or opposite, of a statement. Ex: Boys are trouble Negation: Boys are not trouble A, B, C are collinear Negation: A, B, C are non-collinear Write the converse, inverse, and contrapositive of each conditional statement. 1) If three points are collinear, then they determine a plane. 2) If a segment is bisected, then the segment is cut in half. 3) Students do well on tests when they study.

Lesson 1 Practice: Conditional Statements

Determine if each statement is true or false. T F 1. Points P and S are collinear. T F 2. Points W, R, and V are coplanar. T F 3. QS and TV are coplanar. T F 4. Points Q, R, S, and T are coplanar. T F 5. Line QS lies in plane A. T F 6. Line TV lies in plane A. T F 7. Line QS and line TV intersect only in point R. T F 8. The hypothesis of the conditional statement, “If two angles have the same measure, then they are

congruent” is “two angles have the same measure”. T F 9. The converse of p → q, is q → p. T F 10. If a conditional statement is true, its converse is also true. T F 11. A conditional statement and its contrapositive mean the same thing. T F 12. If a statement is true, its negation is false. Write the negation of each statement. 13. Acute angles are not less than 90°. 14. Chocolate is an ideal food. Write the following conditional statements in the “If…., then…..” form. 15. Adjacent angles share a common side. 16. Perpendicular lines form 4 right angles.

Q

R

S

T V

P

W

A

Write the converse, inverse, and contrapositive of the following conditional statement. Determine if each statement is true or false. 17. If four points are collinear, then they are coplanar. Converse Inverse Contrapositive 18. Find the midpoint of the segment with endpoints at (5, -2) and (-1, 10). 19. If an endpoint of a segment is at (5, -3), and its midpoint is at (1, 2), find the other endpoint of the

segment. 20. Find the distance between the following points. a. (4, -2) and (1, 4) b. (3, 7) and (8, -5) 21. Solve for x and y. x = __________ y = __________

(3x + 40)°

(6x – 35)° (2y + 1)°

Lesson 2: Biconditional Statements and Logical Reasoning Perpendicular lines – lines that intersect to form a right angle Line perpendicular to a plane – a line that intersects a point in the plane and is perpendicular to every line in the plane Biconditional Statement – a statement that contains the phrase “if and only if” Same as writing a conditional statement AND its converse Conditional Statement: If three lines are coplanar, then they lie in the same plane. Converse: Biconditional: Three lines are coplanar if and only if they lie in the same plane. Ex 1) Biconditional: x = 3 if and only if x2 = 9. True or False? Rewrite the biconditional statement as a condtional statement and its converse. Ex 2) The ceiling fan runs if and only if the light switch is on. Conditional: Converse: Ex 3) You scored a touchdown if and only if the football crossed the goal line. Conditional: Converse: Ex 4) The expression 3x + 4 is equal to 10 if and only if x is 2. Conditional: Converse:

Conditional Statement: p q Converse: Inverse: Contrapositive: Inductive Reasoning - examples and patterns are used to form a conjecture Deductive Reasoning – facts, definitions, and postulates are used to write a logical argument Two Laws of Deductive Reasoning: 1) Law of Detachment If p q is a true conditional statement and p is true, then q is true. 2) Law of Syllogism If p q and q r are true conditional statements, then p r is true. Determine if the conclusion is valid. If it is valid, identify which law you used. Ex 1) If the sun is shining, then it is a beautiful day. The sun is shining. Conclusion: It is a beautiful day. Ex 2) If Chris watches a karate movie, then he beats up his little brother. If Chris beats up his little brother, then he gets grounded for a week. Conclusion: If Chris watches a karate movie, then he gets grounded for a week. Ex 3) If Cheryl becomes a nurse, then she will take care of her father. Cheryl takes care of her father. Conclusion: Cheryl became a nurse. Ex 4) If Eric plays too much Call of Duty, then his girlfriend will dump him. If Eric plays too much Call of Duty, then he will lose his job. Conclusion: If Eric’s girlfriend dumps him, then he will lose his job. Ex 5) If Nicole does the dishes alone, then her husband is in trouble. Nicole’s husband is in trouble. Conclusion: Nicole did the dishes alone. Ex 6) If two points are collinear, then they are also coplanar. If two points are coplanar, then they lie in the same plane. Conclusion: If two points are collinear, then they lie in the same plane. Ex 7) If two lines are perpendicular, then they form right angles. Two lines are perpendicular. Conclusion: Right angles are formed Ex 8) If Mr. Leman doesn’t like you, then you will fail geometry. You fail geometry. Conclusion: Mr. Leman doesn’t like you.

Lesson 2 Practice: Logical Reasoning For each of the following determine if the conclusion is valid. If the conclusion is valid, justify it with either the Law of Detachment or the Law of Syllogism. 1. If Nicole graduated from Coal City High School, then she has a diploma. Nicole has a diploma. Conclusion: Nicole graduated from Coal City High School. 2. If Todd is a fox, then he chases chickens. Todd chases chickens. Conclusion: Todd is a fox. 3. If Jake plays Rock Band, then he will learn to play the real guitar. If Jake plays the real guitar, he will get a gorgeous date for homecoming. Conclusion: If Jake plays Rock Band, then he will get a gorgeous date for homecoming. 4. If Brandon studies geometry, then he passes the test. If Brandon passes the test, then he will be happy. Conclusion: If Brandon studies geometry, then he will be happy. 5. If the Scholastic Bowl team plays lots of video games, then they will have quick thumbs. If the Scholastic Bowl team plays lots of video games, then they will lose sleep at night. Conclusion: If the Scholastic Bowl team has quick thumbs, then they will lose sleep at night. 6. If the measure of an angle is less than 90°, then it is acute. m∠A = 60°. Conclusion: ∠A is acute.

Determine a valid conclusion using the Law of Detachment or Law of Syllogism. If a valid conclusion cannot be reached, state that there is no valid conclusion possible. 7. If Donald takes a nap in the back yard, then Chip and Dale will anger him. Chip and Dale anger Donald. Conclusion: 8. If Eric goes to the Joliet, then he shops at Best Buy. If Eric goes to Joliet, then he eats at Taco Bell. Conclusion: 9. If Kirk plays on the tennis team, then he knows how to volley. Kirk knows how to volley. Conclusion: 10. If Bob does not eat school lunch, then he eats gorgonzola cheese. If Bob eats gorgonzola cheese, then his girlfriend won’t kiss him. Conclusion: 11. If three points are noncollinear, then they are contained in a plane. Three points are noncollinear. Conclusion: 12. If a segment is bisected by a line, then the line passes through the midpoint of the segment. If a line passes through the midpoint of a segment, then the two created segments are congruent. Conclusion: 13. If perpendicular lines intersect, then right angles are formed. If perpendicular lines intersect, then four 90o angles are created. Conclusion:

Rewrite the biconditional statement as a conditional statement and its converse. 14. Two angles are congruent if and only if they have the same measure. Statement: Converse: 15. A ray bisects an angle if and only if it divides the angle into two congruent angles. Statement: Converse: 16. Two lines are perpendicular if and only if they intersect to form right angles. Statement: Converse: Determine if each statement is true or false. T F 17. Perpendicular lines intersect to form right angles. T F 18. Inductive reasoning uses patterns and observations to make conjectures. T F 19. The converse of “If this is homecoming week, then we will see all kinds of weird outfits” is “If

we see all kinds of weird outfits, then this is homecoming week.” T F 20. The inverse of “If this is homecoming week, we will see all kinds of weird outfits” is “If this is

not homecoming week, then we will not see all kind of weird outfits.” T F 21. A conditional statement and its contrapositive mean the same thing. T F 22. If two lines intersect, their intersection is a point. T F 23. If two planes intersect, their intersection is a line. T F 24. If two lines intersect, then exactly one plane contains them. T F 25. A biconditional statement is considered true if the converse is true. T F 26. is perpendicular to line m. T F 27. Line n bisects . T F 28. and are supplementary. T F 29. is perpendicular to line p. T F 30. Points A, F, and G are collinear.

A

B C

D

E F

G H J

m

p

n

31. Find the midpoint of the segment that goes from (4, -1) to (-2, 7). 32. If (3, -2) is the midpoint of a line segment and (1, 4) is one endpoint, what is the other endpoint. 33. For A(3, 1) and B(-2, 6), find AB. 34. x = __________ 35. x = __________ 36. ∠VWX is a right angle 37. KM bisects ∠JKL

m∠VWY = (2x-8)° m∠JKM = (6x-6)°

m∠XWY = (x + 50)° m∠MKL = (4x+6)°

x = __________ x = __________

m∠VWY = __________ m∠JKM = __________

m∠XWY = __________

38. An angle is 2 more than 3 times its complement. Find the measure of the angle.

(2x+2) ° (12x+10) ° (11x-4) ° (9x+6) °

W X

Y V J

L K

M

Lesson 3: Reasoning with Properties Algebraic Properties of Equality Let a, b, and c be real numbers: Addition Property: If a = b, then a + c = b + c

Example: x − 3 = 5 + 3 + 3 x = 8

Subtraction Property: If a = b, then a – c = b – c

Example: x + 6 = 10 − 6 − 6 x = 4

Multiplication Property: If a = b, then ac = bc

Example:

x

3= −6

3 ⋅x

3= 3 ⋅ −6

x = −18

Division Property: If a = b and c 0, then

Example: 2x = 12÷2 ÷ 2 x = 6

Substitution Property: If a = b, then a can replace b in any expression or equation

Example: x = yx = 32Therefore, y = 32

3x + 7x = 20y10x = 20y

Distributive Property: a(b + c) = ab + bc

Example: −2(x + 4)−2x − 8

Name the property that justifies each statement. Ex 1) If , then . A) Addition Prop B) Multiplication Prop C) Subtraction Prop Ex 2) If ST = 2 and SU = ST + 3, then SU = 5. A) Addition Prop B) Symmetric Prop C) Substitution Prop Ex 3) If , then A) Symmetric Prop B) Transitive Prop C) Substitution Prop Ex 4) If JK = PQ and PQ = ST, then JK = ST. A) Reflexive Prop B) Symmetric Prop C) Transitive Prop Complete the proof using properties of equality. Given: 5x −18 = 3x + 2 Prove: x = 10 Statements Reasons 1. 5x −18 = 3x + 2 1. Given 2. 2x −18 = 2 2. A) Addition B) Subtraction C) Substitution 3. 2x = 20 3. A) Addition B) Division C) Substitution 4. x = 10 4. A) Addition B) Division C) Multiplication Given: 55z − 3(9z +12) = −64 Prove: z = −1 Statements Reasons 1. 55z − 3(9z +12) = −64 1. Given 2. 55z − 27z − 36 = −64 2. A) Mult. B) Distributive C) Subst. 3. 28z − 36 = −64 3. A) Subtraction B) Addition C) Subst. 4. 28z = −28 4. A) Addition B) Mult. C) Transitive 5. z = −1 5. A) Mult. B) Division C) Addition

Complete the proof using properties of equality. Given: 3(4v −1) − 8v = 17 Prove: v = 5 Statements Reasons 1. 3(4v −1) − 8v = 17 1. 2. 12v − 3− 8v = 17 2. 3. 4v − 3 = 17 3. 4. 4v = 20 4. 5. v = 5 5. Given: AB = CD Prove: AC = BD Statements Reasons 1. AB = CD 1. A) Given B) Prove C) Subst. 2. AB + BC = BC + CD 2. A) Addition B) Symmetric C) Subst. 3. AC = AB + BC 3. A) Seg + Post B) Reflexive C) Subst. 4. BD = BC + CD 4. A) Seg + Post B) Transitive C) Subst. 5. AC = BD 5. A) Symmetric B) Reflexive C) Subst.

A D C B

Given: Prove: m∠4 = 33° Statements Reasons 1. m∠1+ m∠2 = 66° 1. 2. m∠1+ m∠2 + m∠3 = 99° 2. 3. 66° + m∠3 = 99° 3. 4. m∠3 = 33° 4. 5. m∠3 = m∠1,m∠1 = m∠4 5. 6. m∠3 = m∠4 6. 7. m∠4 = 33° 7.

Determine a valid conclusion using the Law of Detachment or Law of Syllogism. Ex 1) If proofs are part of Geometry, then Geometry is fantastic. Proofs are fantastic. Conclusion: Ex 2) If Eric drives the golf cart into the water hazard, he will get thrown off the course. If Eric drives the golf cart into the water hazard, then he will have to pay for the damages. Conclusion: Ex 3) If the two angles add up to 90o, then they are complementary. m∠A + m∠B = 90° Conclusion:

m∠1+ m∠2 = 66°m∠1+ m∠2 + m∠3 = 99°m∠1 = m∠3m∠1 = m∠4

Lesson 3 Practice: Reasoning with Properties

Name the algebraic property that justifies each statement. 1. If x = y + 2 and y + 2 = 12, then x = 12. 2. If x + 3 = 7, then x = 4 3. xy = xy 4. If 7x = 42, then x = 6. 5. If XY – YZ = XM, then XY = XM + YZ 6. 3(x – 4) = 3x – 12 7. If m∠A + m∠B = 90° and m∠B = 30°, then m∠A + 30° = 90°. 8. If m∠A = m∠B, then m∠B = m∠A. 9. If 3x + 2x = 40, then 5x = 40.

10. If , then x – 10 = 12x.

11. If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C. 12. If 4x – 5 = 31, then 4x = 36. 13. m∠A = m∠A 14. If AB + BC = AC and BC = 6, then AB + 6 = AC. 15. If 4x(x – 1) = 12, then 4x2 – 4x = 12. Complete each proof by naming the property that justifies each statement. 16. Given: 2(x – 3) = 8 Prove: x = 7 Statements Reasons 1. 2(x – 3) = 8 1. 2. 2x – 6 = 8 2. 3. 2x = 14 3. 4. x = 7 4.

17. Given:

Prove: x = 4 Statements Reasons

1. 1.

2. 2.

3. 3.

4. x = 4 4. 18. Given: Prove: Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 19. Find the distance between the points (3, 4) and (-2, -1). 20. Find the midpoint of the segment from (6, -1) to (-2, 5). 21. Solve for x and y. x = __________ y = _________ 22. One angle is 18° less than twice its complement. Find the measure of the angle.

(3x + 4)° (5x – 12)° (10y + 12)°

Make a valid conclusion (if possible) for each set of statements using the Law of Detachment or the Law of Syllogism. 23. If Kirby climbs the tree, then he will fall. If Kirby falls, then he will break his foot. Conclusion: 24. If Copper is a hound dog, then he will howl when he finds what he’s been tracking. Copper howled when he found what he was tracking. Conclusion: 25. If two angles are adjacent, then they are not vertical angles. If two angles are not vertical angles, then they are a linear pair. Conclusion: 26. If the basset hound parade is canceled, Mr. Leman will be devastated. The basset hound parade is canceled. Conclusion: Determine if each statement is true or false. T F 27. Perpendicular lines always intersect to form four right angles. T F 28. Deductive reasoning uses patterns and observations to make conjectures. T F 29. The inverse of “If you do not vote, then you cannot complain about the elected” is “If you vote,

then you can complain about the elected.” T F 30. If the inverse of a conditional statement is true, then the converse is also true. T F 31. A biconditional statement is true if both the conditional statement and converse are true. T F 32. If two planes intersect, then they intersect at exactly one point. T F 33. Vertical angles are congruent. T F 34. Vertical angles are never supplementary. T F 35. A biconditional statement is false only if both the conditional statement and converse are false.

Lesson 4: Proving Statements About Segments

Theorem – a statement that follows as a result of other true statements. *MUST BE PROVEN Two – Column Proof: GIVEN: PROVE: Symmetric Property of Congruence: If , then Reflexive Property of Congruence: For any segment AB, Transitive Property of Congruence: If and , then Paragraph Proof: (of Symmetric Property of Congruence) You are given that . By the definition of congruent segments, . By the symmetric property of equality, . Therefore, by the definition of congruent segments, it follows that .

Statements Reasons

1. 2. 3. 4.

1. 2. 3. 4.

P Q

X Y

Ex 1) GIVEN: LK = 5, JK = 5, PROVE: Ex 2) GIVEN: Q is the midpoint of

PROVE:

Statements Reasons

1. LK = 5 2. JK = 5 3. 4. 5. 6.

1. 2. 3. 4. 5. 6.

Statements Reasons

1. Q is the midpoint of 2. PQ = QR 3. 4. 5.

6.

7.

1. 2. 3. 4. 5. 6. 7.

Ex 3) GIVEN: , , PROVE:

Statements Reasons

1. , , 2. 3. 4. 5.

6. 7.

1. 2. 3. 4. 5. 6. 7.

U V W X Y Z

Lesson 4 Practice: Proving Statements about Segments Complete the following proofs. 1. Given: B is the midpoint of AC

BC ≅ DC

Prove: AB ≅ DC

Statements Reasons

1. B is the midpoint of AC 1.

2. AB ≅ BC 2.

3. BC ≅ DC 3.

4. AB ≅ DC 4.

2. Given: AB ≅ AE

BC ≅ ED

Prove: AC ≅ AD

Statements Reasons

1. AB ≅ AE 1.

BC ≅ ED

2. AB = AE 2.

BC = ED

3. AB + BC = AE + ED 3.

4. AB + BC = AC 4.

AE + ED = AD

5. AC = AD 5.

6. AC ≅ AD 6.

3. Write a paragraph proof on a separate sheet of paper.

Given: m∠PMN = m∠RBC

Prove: m∠ABR + m∠PMN = m∠ABC

3. Tell which property is illustrated by each of the following.

a. If 3x + 7 = 40, then 3x = 33.

b. If 6x = 30, then x = 5.

c. If 4(2x + 1) = 20, then 8x + 4 = 20.

d. If 3x + 2y = 24 and y = x + 3, then 3x + 2(x + 3) = 24.

e. If AB + BC = AC, then AB = AC – BC.

f. If m∠A + m∠B = 90° and m∠B = 40°, then m∠A + 40° = 90°.

g. If 45 = 3x + 12, then 3x + 12 = 45.

h. If a = b and b = c, then a = c.

i. If 4x + 2x + 10 = 16, then 6x + 10 = 16.

j. m∠X = m∠X.

4. Determine if each statement is true or false.

T F a. If a statement is true, its negation is false. T F b. The converse of p → q is q → p. T F c. If a conditional statement is true, its converse is also true. T F d. The inverse of p → q is p → ~ q. T F e. The converse of a conditional statement and the inverse of the conditional statement mean

the same thing. T F f. Inductive reasoning uses patterns and observations to make conclusions. T F g. Inductive reasoning is what we use in proofs. 5. Find the midpoint of the segment from (-4, 5) to (6, -9). 6. Find the distance between the points (-4, 5) and (6, 9).

Lesson 5: Proving Statements about Angles Reflexive: For any angle A, ∠A ≅ ∠A.

Symmetric: If ∠A ≅ ∠B, then ∠B ≅ ∠A.

Transitive: If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. Given: ∠1 is a right angle ∠2 is a right angle

Prove: ∠1 ≅ ∠2

Statements Reasons

1. ∠1 is a right angle 1. Given ∠2 is a right angle 2. m∠1 = 90° 2. m∠2 = 90° 3. m∠1 = m∠2 3. 4. ∠1 ≅ ∠2 4.

Theorem: All right angles are congruent. ***Now that we have proven this theorem true, we can use it as a REASON in other proofs.***

1 2

Where does statement 2 come from? We know that all right angles measure 90o because that is how we define and identify a right angle.

Where does statement 3 come from? We need to show that angle 1 and angle 2 have equal measures so that we can say they are congruent and prove statement 4. Why isn’t it substitution? Substitution only replaces one item at a time, for example 2x + 3x can be replaced by 5x. But the transitive property states that two items equal the same thing, for example m∠1 = 90° and m∠2 = 90°, so m∠1 = m∠2

Given: ∠1 and ∠2 are supplements ∠3 and ∠4 are supplements ∠1 ≅ ∠3

Prove: ∠2 ≅ ∠4

Statements Reasons

1. ∠1 and ∠2 are supplements 1. ∠3 and ∠4 are supplements ∠1 ≅ ∠3 2. m∠1 + m∠2 = 180° 2. m∠3 + m∠4 = 180° 3. m∠1 + m∠2 = m∠3 + m∠4 3. 4. m∠1 = m∠3 4. 5. m∠2 = m∠4 5. 6. ∠2 ≅ ∠4 6.

1 2

3 4

Theorem: If two angles are congruent, then their supplements are congruent.

Theorem: If two angles are congruent, then their complements are congruent.

Given: ∠1 and ∠2 are vertical angles

Prove: ∠1 ≅ ∠2

Statements Reasons

1. ∠1 and ∠2 are vertical angles 1. 2. ∠1 and ∠3 are a linear pair 2. ∠2 and ∠3 are a linear pair 3. ∠1 and ∠3 are supplementary 3. ∠2 and ∠3 are supplementary 4. ∠3 ≅ ∠3 4. 5. ∠1 ≅ ∠2 5.

Theorem: Vertical angles are congruent.

Given: ∠3 and ∠4 are vertical angles

Prove: ∠3 ≅ ∠4

Statements Reasons

1 2 1 2 3

3 4

Postulate: If two angles form a linear pair, then they are supplementary.

2

Justify each of the following statements with a definition, postulate, or theorem. ∠2 ≅ ∠3 If BC DC, then ∠ABC is a right angle. If m∠2 = 40° then m∠4 = 140°

Given: m∠1 + m∠2 = 90°

∠2 and ∠3 are complementary angles

Prove: ∠1 ≅ ∠3

Statements Reasons

1. m∠1 + ∠2 = 90° 1. 2. ∠1 and ∠2 are 2. complementary angles 3. ∠2 and ∠3 are 3. complementary angles 4. ∠2 ≅ ∠2 4. 5. ∠1 ≅ ∠3 5.

1 2

3

A B

C D

E

F

2 1 3 4

5 6

1) Given: ∠1 is a right angle ∠2 is a right angle

Prove: ∠1 ≅ ∠2

2)

Given: ∠1 and ∠2 are supplements ∠3 and ∠4 are supplements ∠1 ≅ ∠3

Prove: ∠2 ≅ ∠4

3) Given: ∠1 and ∠2 are vertical angles

Prove: ∠1 ≅ ∠2

1 2

1 2 3 4

1 2 3

Lesson 5 Practice: Proving Statements About Angles

Complete the following proofs: 1. Given: ∠1 is a right angle

∠2 is a right angle

Prove: ∠3 ≅ ∠4

Statements Reasons 1. ∠1 ≅ ∠3 1. 2. ∠1 is a right angle 2. ∠2 is a right angle 3. ∠1 ≅ ∠2 3. 4. ∠2 ≅ ∠3 4. 5. ∠2 ≅ ∠4 5. 6. ∠3 ≅ ∠4 6. 2. Given: ∠1 ≅ ∠4

Prove: ∠2 ≅ ∠3

Statements Reasons 1. ∠1 and ∠2 are a linear pair 1. ∠3 and ∠4 are a linear pair 2. ∠1 and ∠2 are supplementary 2. ∠3 and ∠4 are supplementary 3. ∠1 ≅ ∠4 3. 4. ∠2 ≅ ∠3 4.

1 3

4 2

1 2 3 4

3. Given: ∠2 ≅ ∠3

Prove: ∠1 ≅ ∠4 Statements Reasons

Justify each of the following statements with a definition, postulate, or theorem. 4. If AD ⊥ DC, then ∠6 is a right angle. 5. ∠ 1 ≅ ∠3 6. ∠5 and ∠6 are a linear pair. 7. If ∠7 and ∠8 are complementary, then m∠7 + m∠8 = 90°. 8. If AC bisects ∠DAB, then ∠7 ≅ ∠8. 9. DE + EC = DC 10. If ∠5 is a right angle, then m∠5 = 90°. 11. If ∠7 ≅ ∠2 and ∠2 ≅ ∠4, then ∠7 ≅ ∠4.

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12. If BE bisects DC, then E is the midpoint of DC. 13. If m∠7 + m∠8 = 90° and m∠8 = 50°, then m∠7 + 50° = 90°. 14. ∠2 ≅ ∠2 15. If DE + EC = DC, then DE = DC – EC. 16. m∠7 + m∠8 = m∠DAB 17. If 3(x – 4) + 10 = 16, then 3x – 12 + 10 = 16. 18. If 7x – 2 = 26, then 7x = 28. 19. If 3x + 4x + 12 = 20, then 7x + 12 = 20.

20. If , then 2x + 8 + 1 = 30.

21. If 3x + 5y = 24 and y = x + 2, then 3x + 5(x + 2) = 24. 22. If AB = CD, then CD = AB. 23. If 6x = 42, then x = 7. 24. 24 = 3(x – 5) + 2x, then 3(x – 5) + 2x = 24. True or False. 25. T F The contrapositive of p q is ~q ~p. 26. T F The negation of p is ~p ~q. 27. T F Deductive reasoning uses patterns and observations to make conjectures. 28. T F Vertical angles are never complementary.

For each of the following determine a valid conclusion, if possible, using either the Law of Detachment or the Law of Syllogism. 29. If two angles are supplementary, then their sum is 180o. Conclusion: 30. If two points are collinear, then they are also coplanar. If two points are coplanar, then they must lie in the same plane. Conclusion: 31. If a biconditional statement is false, then either the conditional statement or its converse is false. The biconditional statement is false. Conclusion: 32. If two angles are vertical angles, then they are congruent. If two angles congruent, then they must have the same measure. Conclusion: 33. If the measure if an angle is more than 90°, then it is obtuse. m∠A = 60°. Conclusion: 34. Write the if-then form, converse, inverse, and contrapositive of the statement: All right angles are congruent. If-then: Converse: Inverse: Contrapositive:

Chapter 2 Test Review 1. State the negation of each statement. a. Vertical angles are congruent. b. Monday will not be my birthday. 2. Write the following statements in “If…then” form. a. Geometry students learn to think. b. Congruent angles have the same measures. 3. For the statement “If angles form a linear pair, then the angles are supplementary” write the a. Converse: b. Inverse: c. Contrapostive: T F 4. If a conditional statement is true, its converse if false. T F 5. If a conditional statement is true, its contrapositive is true. T F 6. If the inverse of a conditional statement is false, then the converse is also false. T F 7. If a statement is true, its negation is false. T F 8. If two angles are congruent, their supplements are congruent. T F 9. If two angles form a linear pair, they are supplementary. T F 10. All right angles are congruent. T F 11. If two lines form a right angle, then they are perpendicular. T F 12. Vertical angles are congruent.

Tell which property, postulate, or definition is illustrated in each of the following: 13. If 3x - 4 = 14, then 3x = 18. 14. ∠A ≅ ∠A 15. If a + 3b = 16 and b = a + 4, then a + 3(a + 4) = 16. 16. If 2x + 5 – 7x = 25, then -5x + 5 = 25. 17. If 24 = 3x + 6, then 3x + 6 = 24. 18. If 4(x + 2) = 40, then 4x + 8 = 40. 19. If ∠A ≅ ∠B, then m∠A = m∠B. 20. If x = y and y = 12, then x = 12.

21. If , then x – 1 = 10.

22. If 6x = 48, then x = 8. 23. Given: 37 = 2(3x – 4) + 3x Prove: x = 5 Statements Reasons 1. 37 = 2(3x – 4) + 3x 1. 2. 37 = 6x – 8 + 3x 2. 3. 37 = 9x – 8 3. 4. 45 = 9x 4. 5. 5 = x 5. 6. x = 5 6.

24. Given: AD ⊥ DC

BC ⊥ DC

∠1 ≅ ∠2 Prove: ∠3 ≅ ∠4 Statements Reasons 1. AD ⊥ DC 1.

BC ⊥ DC 2. ∠ADC is a right angle 2.

∠BCD is a right angle

3. m∠ADC = 90° 3.

m∠BCD = 90° 4. m∠3 + m∠1 = m∠ADC 4.

m∠4 + m∠2 = m∠BCD 5. m∠3 + m∠1 = 90° 5.

m∠4 + m∠2 = 90° 6. ∠3 and ∠1 are complementary 6.

∠4 and ∠2 are complementary 7. ∠1 ≅ ∠2 7. 8. ∠3 ≅ ∠4 8.

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A B

C D

E

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25. Given: AE ≅ BE

EC ≅ ED

Prove: AC ≅ BD Statements Reasons 1. AE ≅ BE 1.

EC ≅ ED 2. AE = BE 2.

EC = ED 3. AE + EC = BE + ED 3. 4. AE + EC = AC 4.

BE + ED = BD 5. AC = BD 5. 6. AC ≅ BD 6.

2 3

A B

C D

E

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