Geo metry C o urs e Ou tlin e - Weebly

Geometry - 1 -

G e o m e t r y C o u r s e O u t l i n e

Primary Textbook: Ron Larson, Laurie Boswell, Timothy D. Kanold, and Lee Stiff

Geometry, McDougal Littell, 2007

Unit 1: Essentials of Geometry Describing geometric figures

Measuring geometric figures

Understanding equality and congruence

Unit 2: Reasoning and Proof Use inductive and deductive reasoning

Understanding geometric relationships in diagrams

Writing proofs of geometric relationships

Unit 3: Parallel and Perpendicular Lines Using properties of parallel and perpendicular lines

Proving relationships using angle measures

Making connections to lines in algebra

Unit 4: Congruent Triangles Classifying triangles by sides and angles

Proving that triangles are congruent

Using coordinate geometry to investigate triangle relationships

Unit 5: Relationships within Triangles Using properties of special segments in triangles

Using triangle inequalities to determine what triangles are possible

Extending methods for justifying and proving relationships

Unit 6: Similarity Using ratios and proportions to solve geometry problems

Showing that triangles are similar

Using indirect measurement and similarity

Unit 7: Right Triangles and Trigonometry Using the Pythagorean Theorem and its converse

Using special relationships in right triangles

Using trigonometric ratios to solve right triangles

Unit 8: Quadrilaterals Using angle relationships in polygons

Using properties of parallelograms

Classifying quadrilaterals by their properties

(cont)

Geometry - 2 -

G e o m e t r y C o u r s e O u t l i n e ( c o n t )

Unit 9: Properties of Transformations Performing congruence and similarity transformations

Making real-world connections to symmetry and tessellations

Applying matrices and vectors in Geometry

Unit 10: Properties of Circles Using properties of segments that intersect circles

Applying angle relationships in circles

Using circles in the coordinate plane

Unit 11: Measuring Length and Area Using area formulas for polygons

Relating length, perimeter, and area ratios in similar polygons

Comparing measures for parts of circles and the whole circle

Unit 12: Surface Area and Volume of Solids Exploring solids and their properties

Solving problems using surface area and volume

Connecting similarity to solids

Geometry - 3 -

Geometry Unit 1

Pre-Rev. A Review linear equations and proportions.

Pre-Rev. B Review simplifying radicals and systems of equations.

1. Identify points, lines and planes. (Section 1.1)

2. Use segments and congruence. (Section 1.2)

3. Use midpoint and distance formulas. (Section 1.3)

4. Measure and classify angles. (Section 1.4)

5. Describe angle pair relationships. (Section 1.5)

Review

Geometry - 4 -

Unit 1 Pre-Review A

Algebra Review

Solve the following equations.

1. n – 4 = 9 2. p + 7 = –7 3. 8 + f = –8

4. 4c = 96 5. 3

x = 21 6. –

2

3

x = 6

7. 5n + 4 = 29 8. 8y – 7 = 17 9. 9x – 5 = –14

10. 15 + 5y = 20 11. 5y – 3 – 4y = 5 12. m + 3m + 2 + 2m = 14

13. 5x + 4 – 2x = 2 + 2 14. 3x – 1 = 8x + 9 15. 4y = 2y + 6

16. 4(x + 5) = 28 17. 5(x – 3) + 8 = 18 18. 3(5x – 4) = 8x + 2

Solve the following proportions.

19. 4

3 15

x 20.

88

2x 21.

3

6 12

x x

22. 5

9 6

x x 23.

4 1

5 5x 24.

8

3 9

x x

Geometry Unit 1 Pre-Review B

Radical and Systems of Equations Review

Simplify:

1. 81 2. 24 3. 600

4. 18 5. 36 64 6. 24 28

7. 196 8. 120 9. 27

10. 2 210 5 11. 2 25 1 7 1( ) ( ) 12. 2 22 4 8 4( ) ( )

Solve the following for x and y: 13. y = 3x 14. 4x – 5y = 92 15. x = 8 + 3y

5x + y = 24 x = 7y 2x – 5y = 8

16. 3x + 16 = 100 17. 6x + 12 = 180 18. 4x = 10y

2x = 7y 3x = 4y 7x + 90 = 125

Geometry - 5 -

Worksheet 5A

1. A and B are complementary. A and C are supplementary.

If m C = 100°, determine the measures of A and B.

2. L and M are complementary. M and P are supplementary.

If m L = 20°, determine the measures of M and P.


A B. Determine the measures of A, B and C.

4. The larger of two supplementary angles measures 8 times the smaller. Determine the

measures of the two angles.

5. The measure of an angle is 4 times the measure of its complement. Determine the

measures of both angles.

Unit 1 Worksheet 5B

1. The complement of an angle is five times the measure of the angle itself. Determine the

angle and its complement.

2. Determine the measure of an angle that is 50° more than that of its complement.

3. Determine the measure of an angle that is 60° more than that of its supplement.

4. The supplement of an angle is 30° less than twice the measure of the angle itself.

Determine the angle and its supplement.

5. The supplement of an angle is twice as large as the angle itself. Determine the angle and

its supplement.

6. The complement of an angle is 6° less than twice the measure of the angle itself.

Determine the angle and its complement.

7. Two angles are congruent and complementary. Determine their measures.

8. The complement of an angle is twice as large as the angle itself. Determine the angle and

its complement.

9. The supplement of an angle is 20° more than three times the angle itself. Determine the

angle and its supplement.

10. Determine the measure of an angle that is 18° more than half of its complement.

Geometry - 6 -

Unit 1 Review

In problems 1 – 19 select the correct multiple choice response.

Diagrams are not drawn to scale.

1. Write a name for the given figure

a. EF b. EF

c. FE d. FE

2. Which of the following is not a name for the given figure?

a. AB b. l

c. B d. BA

3. Name the vertex of SRT

a. S b. R

c. T d. cannot determine without a diagram

4. Classify the given angle

a. right angle b. obtuse angle

c. acute angle d. straight angle

5. 1 and 2 are what kind of angles?

a. vertical angles b. congruent angles

c. complementary angles d. adjacent angles

6. Which statement is not true regarding the two angles?

a. A B

b. A and B are complementary angles

c. A and B are vertical angles

d. A and B are acute angles

7. All of the following are names for the angle shown except which one?

a. EFM b. F

c. MFE d. FEM

8. Which angles are vertical angles?

a. 2 and 4 b. 1 and 5

c. 2 and 5 d. 1 and 3

(cont)

E

A

20

feet h

h

A B

l

1 2

45° A

45°

B

E

F

M

1 2

3

5 4

Geometry - 7 -

Review (cont)

9. In the figure, QS bisects PQR. If m 1 = (6x + 18)° and

m 2 = (9x)°, find m PQR

a. 88° b. 105° c. 108°

d. 110° e. 135°

10. XF bisects EXG. If m EXF = 34°, what is the

measure of EXG?

a. 34° b. 68°

c. 56° d. 45°

11. What is another name for angle 2?

a. U b. TUM

c. UKM d. MUK

12. In the figure shown, m 1 = (5x)° m 2 = (6x + 10)° m ABC = 120°

Which equation could be used to find the value of x?

a. 5x = 6x + 10

b. 5x = 120

c. 6x + 10 = 120

d. 5x + 6x + 10 = 120

13. In the figure name three collinear points.

a. S, P, Y b. S, P, T

c. T, P, X d. T, X, Y

14. Which angle is a supplement to TSY?

a. PST b. YSX

c. PSX d. PSY

(cont)

1 2

R

S P

Q

E

X

F

G

1

2

T

K

M

U

1 2

A

C

B

X

T

P

S

Y

X L

P

S

Y T

Geometry - 8 -

D B A

Review (cont)

15. RB bisects ZRL

ZRB = (4x + 16)°

LRB = 88°

Which equation could be used to find the value of x?

a. 4x + 16 = 88

b. 4x + 16 + 88 = 90

c. 2(4x + 16) = 88

d. x = 88 + 4x + 16

16. Which of the following statements is true for the figure shown and using the

segment addition postulate.

a. DB = BA

b. DA = DB + BA

c. B is the midpoint of DA

17. In the figure shown, B is the midpoint of DA

DB = 5x – 1, BA = 4x + 6 DA = 68

Identify all equations below that are true.

a. 5x – 1 = 4x + 6

b. 5x – 1 + 4x + 6 = 68

c. 2(5x – 1) = 68

d. 2(4x + 6) = 68

18. The endpoints of RS are R (5, –1) and S (–9, 13). Find the coordinates of the

midpoint M.

a. (–4, 12) b. (–2, 6) c. (–7, 7) d. (7, 6)

19. Which of the following is the distance formula?

a. 2 2

2 1 2 1x x y y( ) ( ) b. 2 2

2 1 2 1x x y y( ) ( )

c. 2 2

2 2 1 1x y x y( ) ( ) d. 2 2

2 1 2 1x x y y( ) ( )

For problems 20 – 32 answer True or False:

20. ABC is a right angle

21. Point B is between points A and D

22. AB BD

23. ABC DBC

24. Points A, B and D are collinear

25. Point C is between points A and D

(cont)

Z

B L

R

D B A

Use this figure for

problems 20 - 25

B A

C

B D

Geometry - 9 -

Review (cont)

26. If a straight angle is bisected the resulting angles will always be right angles

27. If a right angle is bisected the resulting angles will always be acute angles.

28. If an obtuse angle is bisected the resulting angles will always be obtuse angles.

29. Point B is between points P and S

30. If two angles are obtuse, then they are congruent.

31. If two angles are right angles, then they are congruent.

32. If two angles are right angles, then they are adjacent.

Solve the following problems showing all work.


If m C is 140°, determine the measures of A and B

P B S

Geometry - 10 -

Unit 2 Objective 0

(Note: Diagrams are not drawn to scale.)

A polygon is a closed plane figure with the following properties:

It is formed by three or more line segments called ____________.

Each segment intersects exactly ____ other segments, one at each endpoint.

No two segments with a common ________________are collinear.

1. Draw 3 polygons. 2. Draw 3 figures that are not polygons.

3. This polygon is a convex polygon. Draw three more.

4.

This polygon is a concave polygon (not convex). Draw three more.

Polygons are classified according to the number of sides they have. Name each.

3 sides ____________________ 8 sides ____________________

4 sides ____________________ 9 sides ____________________

5 sides ____________________ 10 sides ___________________

6 sides ____________________ 12 sides ___________________

7 sides ____________________ n sides ___________________

In an _____________________ polygon, all sides are equal.

In an _____________ polygon, all angles in the interior of the polygon are congruent.

A convex polygon that is both equilateral and equiangular is called a ________polygon.

Classify each polygon by the number of sides. Tell whether the polygon is equilateral, equiangular,

regular, or none of these

5. ____________ 6. ____________ 7. _____________ 8. _____________

(cont)

Geometry - 11 -

Unit 2 Objective 0 (cont)

For each regular polygon find the value of x, the length of a side and the perimeter of the polygon.

9. x = _______ 10. x = _______

Length of each side _______ Length of each side _______

Perimeter = _______ Perimeter = _______

Use the coordinate grid for problems 11 and 12. Show all work!

11. Triangle QRS has vertices Q (1,2), R (4, 6), and

S (5,2). What is the perimeter of triangle QRS? (Draw triangle QRS. Find the side lengths using the

distance formula, then find the perimeter.)

SR = _________________ QR = _________________

QS = _________________ Perimeter = ____________

12. Quadrilateral MATH has vertices M (–4, –2), A (–1, 1), T (2, –2) and H (–1, –5).

What is the perimeter of quadrilateral MATH? (Show your work. Use the distance formula)

MA = _________________ AT = _________________

TH = _________________ HM = _________________

Perimeter = ____________

Is quad MATH equilateral? ________ Why or why not?_________

What additional information would be needed to say quad MATH is regular?

_____________________________________________________________

What is another polygon name for quad MATH if it is regular?_____

2x – 6

5x – 27

5x – 1

4x + 3

Geometry - 12 -

Geometry Unit 2

1. Use inductive and deductive reasoning. Give counterexamples to disprove a

statement. (Section 2.1, Section 2.3)

2. Analyze conditional statements. (Section 2.2)

3. Use postulates and diagrams. (Section 2.4)

4. Reason using properties from Algebra. (Section 2.5)

5. Prove statements about segments and angles. (Section 2.6)

6. Prove angle pair relationships. (Section 2.7)

Review

Geometry - 13 -

Unit 2 Worksheet 1

When you reason that what has happened before will happen again, without exception, you are

using inductive reasoning. Inductive reasoning consists of observing data, recognizing patterns

and making decisions based on past experiences. Through observations we are lead to make a

conjecture which may be true or false.

“Deduce” means to reason from known facts. When you reason deductively, you reach a

conclusion using established rules. You start with statements that are considered true and then

show that other statements follow from them.

Inductive Reasoning

Deductive Reasoning

uses past observations uses facts

uses patterns uses definitions

uses unproven statements uses postulates

uses corollaries

uses previous theorems

CANNOT BE USED IN

MATHEMATICAL PROOFS

CAN BE USED IN

MATHEMATICAL PROOFS

Inductive Reasoning - can be Inaccurate because it‟s based

on feelings, observations, patterns

In problems 1 – 10 state whether the reasoning represents deductive or inductive reasoning.

1. Conclusions are based on feelings.

2. Conclusions are based on observing objects.

3. Conclusions are based on definitions.

4. Conclusions are based on proven facts and accepted statements.

5. Conclusions are based on previous patterns.

6. Conclusions are based on suspicions.

7. Conclusions are based on other theorems.

8. Conclusions are based on established laws.

9. The definition of an even number is that it is a number that when divided by 2 has

a remainder of 0. When Sue divided 16 by 2 she got a remainder of 0. Sue

conjectured that 16 is an even number. What type of reasoning did she use?

(cont)

Inductive Reasoning

Deductive Reasoning

Geometry - 14 -

Unit 2 Worksheet 1 (cont)

10. John was told to fill in the sequence 10, 20, 30, ___.

He conjectured that the missing term was 50. What type of reasoning did he use?

10, 20, 30, 50

In problems 11 – 14, select the correct multiple choice response:

11. Which number serves as a counterexample to the statement?

The square of every integer is an even number.

a. 2 b. 6 c. 5 d. 10

12. The table shows an expression evaluated for four different values of x.

Rick concluded that for every x the value of 2x + 5

produces a positive number. Which value of x

serves as a counterexample to prove Rick‟s conclusion

false?

a. –10 b. –2 c. –1 d. 0

13. Which number serves as a counterexample to the statement

All rational numbers can be written as terminating decimals.

a. 1

2 = 0.5 b.

7

4 = 1.75 c.

1

3 = 0.33 d.

9

500 = 0.018

14. Which multiple serves as a counterexample to the statement

If two integers are added together and the sum is even, then the original two

integers are even.

a. 2, 4 b. 1, 3 c. 0, 6 d. –2, – 4

x 2x + 5

-2 1

0 5

1 7

4 13

add

Geometry - 15 -

Unit 2 Worksheet 2

NOTE: Diagrams are not drawn to scale.

Decide whether the statement is true or false. If false, provide a counterexample.

1. If it is a weekend day, then it is Saturday.

2. If an angle is acute, then its measure is less than 90°.

3. If A B , then m A m B

4. If a = b, then a + c = b + c

5. If a figure is a rectangle, then it has 4 sides.

6. If n > 5, then n > 7.

Write the converse for the statements below and determine if the converse is true or false. If

false, provide a counterexample.

7. If I have 2 dimes and 1 nickel, then I have 25 cents.

8. If 1m = 90°, then 1 is a right angle.

9. If x = – 6, then x2 = 36.

10. If you can divide a number by 4, then you can divide the number by 2.

Select the correct multiple choice response:

11. If two angles share a common vertex, then they are adjacent

Which of the following serves as a counterexample to the assertion above?

a. b.

c. d.

(cont)

1 2 2 1

1

2 1 2

Geometry - 16 -


12. If two lines are coplanar, then they intersect.


a. b. c.

13. A pair of supplementary angles are adjacent to each other.


a. b. c.

14. The definition of congruent segments is:

If two line segments have the same length then they are congruent segments.

a. Write the converse of this definition

b. Write the definition as a biconditional

15. The definition of perpendicular lines is:

If two lines intersect to form a right angle, then they are perpendicular lines.

a. Write the converse of this definition

b. Write the definition as a biconditional

40° 140° 90°

90°

60°

120°

m p

Geometry - 17 -

Unit 2 Worksheet 4 NOTE: Diagrams are not drawn to scale. Name the property illustrated below:

1. P P

2. If AB CD and CD EF , then AB EF

3. If RS = TW, then TW = RS

4. If x + 5 = 16, then x = 11

5. If 5y = – 20, then y = – 4

6. 2(a + b) = 2a + 2b

7. If 2x + y = 70 and y = 3x, then 2x + 3x = 70

8. If AB = CD and CD = 23, then AB = 23

9. Justify each step:

2x + 3 = 11 Given

a. 2x = 8 ______________________

b. x = 4 ______________________ 10. Justify each step:

3

4x = 6 + 2x Given

a. 3x = 4(6 + 2x) ______________________

b. 3x = 24 + 8x ______________________

c. – 5x = 24 ______________________

d. x = – 24

5 ______________________

11. Justify each step:

Given: m AOC m BOD

Prove: 1 3m m

Statements Reasons

1. m AOC m BOD 1. ______________________

2. 1 2m m m AOC 2. ______________________

2 3m m m BOD

3. 1 2m m = 2 3m m 3. ______________________

4. 2 2m m 4. ______________________

5. 1 3m m 5. ______________________

(cont)

1 2 3

A B

C

O D

Geometry - 18 -


12. Given: FL = AT

Prove: FA = LT

Statements Reasons

1. FL = AT 1. ______________________

2. LA = LA 2. ______________________

3. FL LA AT LA 3. ______________________

4. FL LA FA 4. ______________________

AT LA LT

5. FA = LT 5. ______________________

13. Given: RT and PQ intersect at S so that

RS = PS and ST = SQ

Prove: RT = PQ

Statements Reasons

1. RS = PS and ST = SQ 1. ______________________

2. ST = ST 2. ______________________

3. RS ST PS ST 3. ______________________

4. RS ST PS SQ 4. ______________________

5. RS ST RT 5. ______________________

PS SQ PQ

6. RT = PQ 6. ______________________

(cont)

F L A T

R

S

P

Q T

Geometry - 19 -


14. Given: DW = ON

Prove: DO = WN

Statements Reasons

1. DW = ON 1. ______________________

2. DW DO OW 2. ______________________

ON WN OW

3. DO OW = WN OW 3. ______________________

4. OW OW 4. ______________________

5. DO = WN 5. ______________________

15. Given: 1 2m m and 3 4m m

Prove: m PRT m QTR

Statements Reasons

1. 1 2m m and 3 4m m 1. ______________________

2. 3 3m m 2.______________________

3. 1 3 2 3m m m m 3. ______________________

4. 1 3 2 4m m m m 4. ______________________

5. 1 3m m m PRT 5. ______________________

2 4m m m QTR

6. m PRT m QTR 6. ______________________

D O

W N

R T

P Q

1 2 3 4

Geometry - 20 -

Unit 2

Simple Proofs Worksheet 5A

Identify the property used. Choose from this list of justifications.

Addition, Subtraction, Multiplication, Division, Substitution, Reflexive, Symmetric, Transitive

Statement Reason Statement Reason

1. 3a = 6 Given 12.

222

.

b = 3 Given

a = 2 3 = b

2. a + 2 = 5 Given 13. AB = 7 Given

5 = a + 2 AB + 2 = 9

3. CD + DE = CE Given 14. a = 3 Given

CD = CE – DE a – 5 = – 2

4. b = 5 Given 15. AB = 10 Given

3b = 15 2AB = 20

5. 2y = 50 Given 16. xy = 10 Given

y = 25

2

xy = 5

6. AB = CD Given 17. XY + YZ = 15 and 15 = WV Given

AB + 5 = CD + 5 XY + YZ = WV

7. AB = 3 and XY = 3 18. RO = NP Given

AB = XY 3RO = 3NP

8. 3QX = 10 Given 19. 15 – AB = DE Given

QX =

13

3

15 = DE + AB

9. AB + CD = 30 and CD = 10 Given 20. AB + 10 = 30 Given

AB + 10 = 30 AB = 20

10.

0.

AB = 5 and CD = AB Given 21. a < b and b < c Given

CD = 5 a < c

11. a + b > c and c = 10 Given 22. a < b Given

a + b > 10 2a < 2b

Geometry - 21 -

Unit 2 Worksheet 5B


In problems 1 – 14 name the property illustrated.

1. If m = k and k = d, then m = d

2. 3 3m m

3. If 5 = 4 + 1, then 4 + 1 = 5

4. If Y is between X and Z, then XY + YZ = XZ

5. If x + 7 = 9, then x = 2

6. If point A is in the interior of XYZ , then 1 2m m m XYZ

.

7. a(b + c) = ab + ac

8. If m + d = 7 and d = k, then m + k = 7

9. If m A m B = 90°, then A and B are complements.

10. 1 2m m = 180°

11. If y – 5 = 11, then y = 16

12. If P is between A and B and AP = PB. then P is the midpoint of AB

13. If m XYW m WYZ , then YW

bisects XYZ

14. If WY = KD and KD = AB, then WY = AB

15. If B is between A and C and AB = BC, what do you call point B?

16. Write a counterexample to the statement:

If xy = 6, then x = 2 and y = 3.

Z Y X

B P A

A

X

Y Z

1 2

W

X

Y Z

Geometry - 22 -

Unit 2 Worksheet 6


1. Given: 1 3m m

Prove: m DEG m HEF

Statements Reasons

1. 1 3m m 1.

2. 2 2m m 2.

3. 1 2 3 2m m m m 3.

4. 1 2m m m DEG 4.

3 2m m m HEF

5. m DEG m HEF 5.

2. Given: KP = ST PR = TV

Prove: KR = SV

Statements Reasons

1. 1.

2. PR = PR 2.

3. KP + PR = ST + PR 3.

4. KP + PR = ST + TV 4.

5. KP + PR = KR 5.

ST + TV = SV

6. KR = SV 6.

3. Given: 1 4m m

Prove: 2 3m m

Statements Reasons

1. 1 4m m 1.

2. 1 2 180m m 2.

3 4 180m m

3. 180° = 180° 3.

4. 1 2 3 4m m m m 4.

5. 1 2 3 1m m m m 5.

6. 2 3m m 6.

(cont)

1 2

3

D E

G

F

H

K P R

S T V

1 2 3 4

Geometry - 23 -


4. Given: m AOI m EOU

Prove: 1 3m m

Statements Reasons

1. m AOI m EOU 1.

2. 1 2m m m AOI 2.

2 3m m m EOU

3. 1 2 2 3m m m m 3.

4. 2 2m m 4.

5. 1 3m m 5.

5. Given: M is the midpoint of PQ

N is the midpoint of RS

PQ = RS

Prove: PM = RN

Statements Reasons

1. M is the midpoint of PQ 1.

N is the midpoint of RS

2. PM = 1

2PQ 2.

RN = 1

2RS

3. PQ = RS 3. Given

4. 1

2PQ =

1

2RS 4.

5. PM = RN 5.

(cont)

1 2

3

A

O

I

U

E

P M Q

R N S

Geometry - 24 -


6. Given: SV

bisects RST

RU

bisects SRT

m RST m SRT

Prove: 1 2m m

Statements Reasons

1. SV

bisects RST 1.

RU

bisects SRT

2. 1

12

m m RST 2.

1

22

m m SRT

3. m RST m SRT 3. Given

4. 1 1

2 2m RST m SRT 4.

5. 1 2m m 5.

7. Given: BA AC

1 is complementary to 3

Prove: 2 3m m

Statements Reasons

1. BA AC 1.

2. BAC is a right angle 2.

3. 90m BAC 3.

4. 1 2m BAC m m 4.

5. 1 2 90m m 5.

6. 1 and 2 are complementary 6.

7. 1 and 3 are complementary 7. Given

8. 1 3 90m m 8.

9. 1 2 1 3m m m m 9.

10. 1 1m m 10.

11. 2 3m m 11.

(cont)

T T

R

V

S

U

1 2

B

A

C D

3

2 1

Geometry - 25 -


8. Given: BD

bisects ABE

Prove: 2 4

Statements Reasons

1. BD

bisects ABE 1.

2. ___ ___ 2. Definition angle bisector

3. 1 4 3. Vertical angles are

4. 2 4 4.

9. Given: 1 2m m

Prove: 4 is supplementary to 5

Statements Reasons

1. 1 2m m 1.

2. 1 5m m 2.

3. 2 5m m 3.

4. 2 4 180m m 4.

5. 5 4 180m m 5.

6. 5 and 4 are supplementary 6.

10. Given: AC = BD

Prove: AB = CD

Statements Reasons 1. AC = BD 1.

2. AB + BC = AC 2.

BC + CD = BD

3. AB + BC = BC + CD 3.

4. BC = BC 4.

5. AB = CD 5.

(cont)

2

C

B

D A

E 3 4

1

2 4

3 5

1

A B C D

Geometry - 26 -


11. Given: 1 3m m

Prove: m PXR m SXQ

Statements Reasons

1. 1 3m m 1.

2. 2 2m m 2.

3. 1 2 3 2m m m m 3.

4. 2 3m m m PXR 4.

1 2m m m SXQ

5. m PXR m SXQ 5.

12. Given: m AXC m DYF , 1 3m m

Prove: 2 4m m

Statements Reasons

1. m AXC m DYF , 1 3m m 1.

2. 1 2m AXC m m 2.

3 4m DYF m m

3. 1 2 3 4m m m m 3.

4. 3 2 3 4m m m m 4.

5. 3 3m m 5.

6. 2 4m m 6.

13. Given: AC = DF, AB = DE

Prove: BC = EF

Statements Reasons

1. AC = DF 1.

2. AC = AB + BC 2.

DF = DE + EF

3. AB + BC = DE + EF 3.

4. AB = DE 4. Given

5. AB + BC = AB + EF 5.

6. AB = AB 6.

7. BC = EF 7.

(cont)

1

P

X R

S

Q

2

3

A 2

1

B C

X

D 4

3

E F

Y

A B C

D E F

Geometry - 27 -


14. Given: 5

3

x = 15

Prove: x = 9

Statements Reasons

1. 1.

2. 5x = 45 2.

3. x = 9 3.

15. Given: 90m TUV 90m XWV

1 3m m

Prove: 2 4m m

Statements Reasons

1. 90m TUV 90m XWV 1.

2. m TUV m XWV 2.

3. 1 2m TUV m m 3.

3 4m XWV m m

4. 1 2 3 4m m m m 4.

5. 1 3m m 5. Given

6. 1 2 1 4m m m m 6.

7. 2 4m m 7.

1

T

V U

X

W 2

3 4

Geometry - 28 -

Unit 2 Review

In problems 1 - 13 select the correct multiple choice response.


1. Which of the following is a true statement?

a. Parallel lines always intersect.

b. Intersecting lines are never parallel

c. Perpendicular lines never intersect.

d. Intersecting lines are always perpendicular.

2. In the figure, m CDE = 80°. CDE is bisected by AB

.

What is the measure of ADC ?

a. 40° b. 180°

c. 220° d. 140°

3. m MPR = 20°, NP PR

Find m LPN

a. 20° b. 160°

c. 90° d. 110°

4. If KA BT , which of the following must be true?

a. KB AT

b. B is the midpoint of KA

c. A is the midpoint of BT

d. A is the midpoint of KT

5. The intersection of Plane R with Plane P is

a. Point K b. AF

c. TC d. TC e. WT

6. Which of the following is a counterexample to the

statement All real numbers have a reciprocal.

a. 5 b. 0

c. 1 d. –7

7. Which of the following is the name for reasoning that uses facts, definitions,

accepted properties and theorems.

a. Logical reasoning

b. Inductive reasoning

c. Deductive reasoning

d. Observational reasoning

(cont)

B C

D

E A

M

N

P R

L

K B A T

M

P

R

A

F

K

C

T

W

Geometry - 29 -

Unit 2 Review (cont)

8. Identify the hypothesis in the following statement:

If the sum of two angles is 90°, then the angles are complementary.

a. There are two angles

b. If the sum of two angles is 90°

c. Then the angles are complementary

d. Angles are complementary if their sum is 90°

9. Which diagram shows two angles that are supplementary and adjacent?

a. b.

c. d.

10. The pair of angles AOB and BOC can best be classified as _____

a. supplementary angles

b. vertical angles

c. adjacent angles

d. right angles

11. Linear angles are always ______

a. obtuse

b. complementary

c. supplementary

d. right

12. Linear angles are NEVER _______

a. vertical

b. adjacent

c. congruent

d. supplementary

13. If two angles are complementary, then the sum of their degree measures is

a. 45°

b. 60°

c. 90°

d. 180°

(cont)

120°

30°

30°

135° 45°

40° 140° 30°

150°

O

C B

A

Geometry - 30 -


In problems 14 - 21 name the property or definition illustrated.

14. If 6x – 7 = 29, then 6x = 36

15. 3(x + y) = 3x + 3y

16. If m A m B and m B m C then m A m C

17. AC = AC

18. If A is a right angle, then m A = 90°

19. If M is the midpoint of AB, then AM = MB

20. If m X m Y = 90°, then X and Y are complementary

21. Given: x + y = 14 and x = 5, then 5 + y = 14

Use this diagram to answer questions 22 – 24.

22. If 3m = 37°, calculate m BDC

23. If m BEC = 88°, calculate m CED and m AED

24. If E is the midpoint of AC and AC = 28, calculate AE

25. Determine the value of x and m ABC

26. Determine the value of x, m AED and m AEB

27. Use the diagram to find the values of x and y

28. The complement of an angle is three more than twice the measure of the angle

itself. Find the measure of the angle and the complement.

Write the following definitions as biconditionals.

29. If points are collinear, then they all lie in one line.

30. If points lie in one plane, then they are coplanar

(cont)

B

D

C

A 3

E

(3x + 28)°

(4y + 38)°

(4y – 2)°

(7x + 12)°

A

(6x + 2)°

(10x – 8)°

B

C

D

E (6x – 14)° (4x + 12)°

A

C

B

D

Geometry - 31 -


31. Complete the proof by filling in the reasons.

Given: 6(x – 4) = x + 16

Prove: x = 8

Statements Reasons

1. 6(x – 4) = x + 16 1.___________________

2. 6x – 24 = x + 16 2.___________________

3. 5x – 24 = 16 3.___________________

4. 5x = 40 4.___________________

5. x = 8 5.___________________

32. Justify the reasons on the following proof:

Given: MO = LD

Prove: ML = OD

Statements Reasons

1. MO = LD 1.___________________

2. OL = OL 2.___________________

3. MO + OL = LD + OL 3.___________________

4. MO + OL = ML 4.___________________

LD + OL = OD

5. ML = OD 5.___________________

Use the diagram for problems 33 - 38.

m DOF = 90°, m BOA = 40°, m GOF = 60°

33. Calculate m BOD

34. Calculate m FOE

35. Calculate m GOE

36. Calculate m BOC

37. Calculate m COD

38. Calculate m DOE

39. Calculate m AOD

40. Calculate m BOG .

D L

O M

G

B

A

C D

E

F H

O

Geometry - 32 -

Unit 2 Geometry Properties Review Directions: Match the name of the properties in the left column with the definitions in the right column.

1. Segment Addition Postulate (pg 10) a) Two angles whose sum is 180°

2. Definition Midpoint (pg 15) b) a = a 3. Angle Addition Postulate (pg 25) c) If a = b, then a can be replaced with b

4. Definition Right Angle (pg 25) d) Two angles whose sum is 90° 5. Congruent Angles (pg 26) e) Two adjacent angles that are supplementary

6. Angle Bisector (pg 28) f) Two angles that share a common vertex and

side, but have no common interior points

7. Defn. Complementary Angles (pg 35) g) If B is between A and C, then AB + BC = AC

8. Defn. Supplementary Angles (pg 35) h) If a = b, then a – c = b – c

9. Adjacent Angles (pg 35) i) Two angles whose sides form two pairs of

opposite rays. The angles are congruent.

10. Definition of Perpendicular Lines (pg

81) j) If a = b, then

a

c =

b

c, ( c 0)

11. Addition Property of Equality (pg 105) k) A ray that divides an angle into two congruent

angles

12. Subtraction Prop.of Equality (pg 105) l) Two lines that form a right angle

13. Mult. Prop. of Equality (pg 105) m) If a = b, then b = a

14. Division Property of Equality (pg 105) n) An angle whose measure is 90°

15. Substitution Prop. of Equality (pg 105) o) If a = b, then a c = b c

16. Distributive Property (pg 106) p) Two angles that have the same measure

17. Reflexive Property of Equality (pg 107) q) If P is in the interior of RST , then

m RST m RSP m PST

18. Symmetric Prop. of Equality (pg 107) r) a (b + c) = ab + ac

19. Transitive Property of Equality (pg 107) s) M is on AB and AM = MB,

20. Linear Pair Postulate (pg 126) t) If a = b, then a + c = b + c

21. Vert. Angles Congruence Thm. (pg 126)

u) If a = b and b = c, then a = c

Geometry - 33 -

Unit 2 Complements and Supplements

Set up an equation for each problem, then solve for x. Use your answer for „x‟ to determine the

angle measures for the problem.

1. The complement of an angle is five times the measure of the angle itself. Determine

the angle and its complement.

2. The supplement of an angle is 30° less than twice the measure of the angle itself.

Determine the angle and its supplement.

3. The supplement of an angle is twice as large as the angle itself. Determine the angle

and its supplement.

4. The complement of an angle is 6° less than twice the measure of the angle itself.

Determine the angle and its complement.

5. Three times the measure of the supplement of an angle is equal to eight times

the measure of its complement. Determine the angle, its complement, and its

supplement.

6. Two angles are congruent and complementary. Determine their measures.

7. Two angles are congruent and supplementary. Determine their measures.

8. The complement of an angle is twice as large as the angle itself. Determine the angle

and its complement.

9. The complement of an angle is 10° less than the angle itself. Determine the angle and

its complement.

10. The supplement of an angle is 20° more than three times the angle itself. Determine

the angle and its supplement.

Geometry - 34 -

Unit 2 Bisectors, Midpoints, Complements, Supplements, Vertical Angles

1. OB bisects AOC

1m = 2x + 20 2m = 5x + 5

a. Calculate the value of x

b. Calculate 1m

c. Calculate m AOC

2. ON bisects MOP

2m = 3x – 7 3m = 2x + 5

a. Determine the value of x

b. Determine 2m

c. Determine 3m

3. OY bisects VOZ

3m = 5x + 2 m VOZ = 12x – 4

a. Determine the value of x

b. Determine 3m

c. Determine m VOZ

4. M is the midpoint of LN

LM = 4 + 3x MN = 7

Calculate the value of x

5. C is the midpoint of BD

BC = 3x – 4 CD = 17

Find the value of x

6. J is the midpoint of HK

HK = 40 JK = 2x + 8

Find the value of x

7. m FGD = 18x + 6 m DGH = 10x

a. Find the value of x

b. Find m FGD

c. Find m DGH

(cont)

A B C

D

E O

1 2 3

4

L

N

P

M

R

O 1

2 3

4

U

W

V

Y

1

Z

0 2 3

4

L

M N

O

P

B A

C D E

K J H

G F

E H

F

G

D

Geometry - 35 -

Unit 2 Bisectors, Midpoints, Complements, Supplements, Vertical Angles (cont)

8. Are X and Y complementary? Explain why or why not.

9. Are 1 and 2 complementary? Explain why or why not.

10. Are M and N complementary? Explain why or why not.

11. Calculate the supplement of C if m C = 35°

12. m AEB = 140°, Determine the measures of the remaining angles.

a. m BEC

b. m CED

c. m DEA

13. Two complementary angles are congruent. Determine their measures. Show algebraic

work.

14. Two supplementary angles are congruent. Determine their measures. Show algebraic

work.

In the diagram AFB is a right angle.

15. Name another right angle.

16. Name two complementary angles.

17. Name two congruent supplementary angles.

18. Name two non-congruent supplementary angles.

19. Name two acute vertical angles.

20. Name two obtuse vertical angles.

(cont)

X

43° 47°

Y

2 1

P

M

N

37°

53°

B

E

D C

A

B

C

D

E

F

A

2 1

Geometry - 36 -

Unit 2 Bisectors, Midpoints, Complements, Supplements, Vertical Angles (cont)

In the diagram, OT bisects SOU ,

m UOV = 35°, m YOW = 120°.

Find the measure of each angle below.

21. m ZOY 22. m ZOW

23. m VOW 24. m SOU

25. m TOU 26. m ZOT

Determine the value of x. Show algebraic work.

27. 28. 29.

U

W

V

S

Z

Y

X O

T

70°

(3x–5)°

(6x–22)°

(3x+8)°

36°

64° 4x°

Geometry - 37 -

Unit 2 Angle Pairs

Use the diagram to decide whether the statement is true or false.

1. If 1m = 47°, then 2m = 43°

2. If 1m = 47°, then 3m = 47°

3. 1 3 2 4m m m m

4. 1 4 2 3m m m m

Make a sketch of the given information. Label all angles which can be determined.

5. Adjacent complementary angles 6. Nonadjacent supplementary angles

where one angle measures 42° where one angle measures 42°

7. Congruent linear pairs 8. Vertical angles which measure 42°

9. ABC and CBD are adjacent 10. 1 and 2 are complementary

complementary angles. CBD 3 and 4 are complementary

and DBF are adjacent 1 and 3 are vertical angles.

complementary angles.

Determine the value of x and y . Calculate the measure of each angle in the diagram.

11. 12.

13. 14.

2

4 3

1

2(3y – 25)°

(4y + 2)°

(13x + 9)°

(15x – 1)°

A

E B

C

D 13x°

2(y + 25)°

(4x + 10)°

(2y – 30)°

A

E B

C

D

(17y – 9)°

(21x – 3)°

4y°

(5x + 1)°

A

E B

C

D

13y

(16y – 27)°

7x°

(5x + 18)°

A

E B

C

D

Geometry - 38 -

Unit 2 Special Pairs of Angles

Determine the measures of a complement and a supplement of C

m C complement supplement C complement supplement

1. m C = 36° 5. m C = 5°

2. m C = 70° 6. m C = 29°

3. m C = 49.2° 7. m C = x°

4. m C = 11° 8. m C = 2x°

In the diagram, m RIE = 90°

9. RIV is complementary to ________.

10. RID is supplementary to ________.

11. DIR is adjacent to angle ________.

12. If m VIN = 80° and m VIE = 32°, then m NIE = _______

In the diagram, m DOF = 90°, m BOA = 30°, and m GOF = 50°

Calculate the measures of the following angles.

13. m BOD = ________ 14. m FOE = ________

15. m GOE = ________ 16. m BOC = ________

17. m COD = ________ 18. m DOE = ________

19. m AOD = ________ 20. m BOG = ________

Calculate the value of x

21. 22.

If C and D are complementary, complete the following.

23. m C = 3x, m D = x – 6 24. m C = x + 10, m D = 2x – 7

x = ___, m C = ___, m D = ___ x = ___, m C = ___, m D = ___

If E and F are supplementary, complete the following.

25. m E = 5y – 3, m F = 2y + 1 26. m E = y – 9, m F = 4y + 14

y = ___, m E = ___, m F = ___ y = ___, m E = ___, m F = ___

A D

B C D

D

E D

F H

G

O 50°

30°

85°

(2x–3)°

(6x+2)°

(3x+71)°

V

D N I

R E

Geometry - 39 -

Unit 3 Objective 0

Perimeter, Area, and Volume Review


Find the perimeter and area of the figures in problems 1-4. 1. 2. 3.

4. 5. A triangle has a base of 33 yards a height

of 56 yards. Sketch the triangle and find

it‟s area.

In problems 6-8 use the information about the figure to find the indicated measure.

6. Perimeter = 84 ft. 7. Area = 432 m2

Find the length, L Find the width, w

8. Area of shaded triangle = 189 cm2

Find the height, h

9. The area of a rectangle is 551 square inches, and its width is 19 inches. Find

the length.

Find the volume of the figures below.

10. 11.

(cont)

32

4 m

13.5 cm

6.9 cm

13 in.

12 in.

5 in.

72 in. 16 in.

34 in. 30in. 78 in.

L

13 ft 24 m

w

15 cm 21 cm

h

4 in.

4 in.

4 in.

2 cm 3 cm

5 cm

Geometry - 40 -

Unit 3 Objective 0 (cont)

12. A game board is made up of 9 squares put into 3 rows and 3 columns as shown.

Each of the 9 squares has sides that measure 5 cm. Find the perimeter of the

game board.

The four sides of the figures below will be folded up and taped to make an open

box. What will be the volume of each box?

13. 14.

15. When the box below is closed it has a length of 7 inches, a width of 4 inches and a

height of 5 inches. What is the volume of the box.

5 cm

5 cm

Geometry - 41 -

Geometry Unit 3

1. Identify pairs of lines and angles. (Section 3.1)

2. Use parallel lines and transversals. (Section 3.2)

3. Prove lines are parallel. (Section 3.3)

4. Two column proofs using parallel line theorems. (Section 3.2 and 3.3)

5. Find and use slopes of lines. (Section 3.4)

6. Prove theorems about perpendicular lines. (Section 3.6)

Review

Geometry - 42 -

Unit 3 Worksheet 3


Use the information given to name the lines that must be parallel. If there are no such segments,

write „none‟

1. m 1 = m 2 2. m 3 = m 4 3. m 5 + m 6 = 180°

4. m 7 = m 8 5. m 9 = m 10 6. m 11 = m 12

7. m 13 = m 14 8. m 1 = m 2 = m 3 9. m 4 = m 5 = m 6

10. m 7 = m 8 11. m 9 + m 10 = 180° 12. m 1 = m 2 = m 3

b

c

d

a

7

8

b

c

d

a

6

5

b

c

d

a

2 1 4

3

b

c

d

a

9

b

c

d

a

10

b

c

d

a

12

11

b

c

d

a

14

13

h k

n

m

f

p

3 2 1

h k

n

m

f

p 6

5 4

h k

n

m

f

p 7

8

h k

n

m

f

p

9 10

h k

n

m

f

p 3

2

1

Geometry - 43 -

Unit 3 Worksheet 4


1. Given: r s , c r

Prove: c s

Statements Reasons 1. 1. Given

2. 1 is a right angle 2.

3. m 1 = 90° 3.

4. 1 2 4.

5. 90° = m 2 5.

6. 2 is a right angle 6.

7. c s 7.

2. Given: a b , 1 3

Prove: 2 3

Statements Reasons

1. 1.

2. 1 2 2.

3. 2 3 3.

3. Given: k m

Prove: 3 is supplementary to 8

Statements Reasons

1. 1.

2. 3 5 2.

3. m 5 m 8 = 180° 3.

4. m 3 m 8 = 180° 4.

5. 3 is supplementary to 8 5.

(cont)

a

b

c 3

1

2

1 2

t

m

k

5

4 3

6

7 8

c

s

r 1

2

Geometry - 44 -


4. Given: a b , 1 3

Prove: c d

Statements Reasons 1. 1.

2. 1 2 2.

3. 2 3 3.

4. c d 4.

5. Given: 1 2 , 3 4

Prove: EL KT

Statements Reasons

1. 1.

2. 2 3 2.

3. 1 4 3.

4. EL KT 4.

6. Given: m 1 m 2 = 90° m 3 m 4 = 90°

Prove: FW GL

Statements Reasons

1. 1.

2. m 1 m 2 = m 3 m 4 2.

3. m 2 m 3 3.

4. m 1 m 4 4.

5. FW GL 5.

(cont)

1

2

d

3

b

a

c

2 4 3

1

E

L

K

M

T

G

4

3

2

1

F

L

M W

Geometry - 45 -


7. Given: BD BA

2 and 3 are complementary

Prove: AC BD

Statements Reasons

1. 2 and 3 are complementary 1. Given

2. m 2 m 3 = 90° 2.

3. BD BA 3. Given

4. m ABD = 90° 4.

5. m 1 m 2 = m ABD 5.

6. m 1 m 2 = 90° 6.

7. m 1 m 2 = m 2 m 3 7.

8. m 1 m 3 8.

9. AC BD 9.

8. Given: AV EV , 1 2

Prove: DE EV

Statements Reasons

1. 1 2 1.

2. AV ED 2.

3. DEV and AVE 3.

are supplements

4. m DEV m AVE = 180° 4.

5. AV EV 5. Given

6. m AVE = 90° 6.

7. m DEV + 90° = 180° 7.

8. m DEV = 90° 8.

9. DEV is a right angle 9.

10. DE EV 10.

2

A

C

B

D

1

3

D

A

V E

1 2

Geometry - 46 -

Unit 3 Worksheet 5

Formulas from Algebra and Geometry can be used to prove a polygon is a particular shape. Use the

distance formula and the formula for the slope of a line for the following coordinate proofs.

DISTANCE = 22 )()( yyxx

SLOPE = changehorizontal

angeverticalch =

12

12

xx

yy =

X

Y

PARALLEL LINES have the same slope.

PERPENDICULAR LINES have slopes that are opposite reciprocals.

Example: A (2, -1) B (-2, 4)

AB = 22 )41()22( = 254 = 29

Slope of AB = 22

)1(4 =

4

5

Slope of any line parallel to AB is 4

5

Slope of any line perpendicular to AB is 5

4

A quadrilateral is a polygon with 4 sides.

A parallelogram is a quadrilateral with opposite sides parallel.

A rectangle is a parallelogram with four right angles. (Sides are perpendicular)

A rhombus is a parallelogram with four congruent sides.

A square is a parallelogram with four congruent sides and four right angles.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

For each problem,

Plot the points and connect in order to form a quadrilateral.

Find the length (distance) of each of the four sides, show your work and organize it neatly!

Find the slope of each of the four sides, show your work neatly!!

State the facts (which sides are congruent, which sides are parallel, which sides are perpendicular) to

prove what type of polygon the shape is, use as many of the above names as fit.

1. A (2, 3), B (5, 1), C (2, –1), D (–1, 1)

2. N(–4, 1), E (–1, 3 ), A( 3, –3 ), T( 0, –5 )

(cont)

Geometry - 47 -


3.a. Find the slopes of SC and CB 4. Given: Quad. LAMB as shown.

b. Calculate the product of their slopes a. Find the slopes of LA and MB

c. Is SC CB ? Justify your answer. b. Is LA MB Justify your answer.

5. Determine if AB CD 6. Given: Quad. CAGE as sketched.

Justify your answer Justify all answers

a. Determine if CA AG

b. Determine if AG GE

c. Determine the length of CE

d. Determine the length of EG

R S

C

B

A L

A

B

M

D

A

B

C

G

A

C

E

Geometry - 48 -

Unit 3 Worksheet 6

1. Given: The slope of line m is 4

7. line m line p

Which statement below must be true?

A. The slope of line p is 4

7

B. The slope of line p is 4

7

C. The slope of line p is 7

4

D. The slope of line p is 7

4

2. Which statement would prove that KF TV

A. (the length of KF) = (the length of TV)

B. (the slope of KF) = (the slope of TV)

C. (the slope of KF) = 1

the slope of TV

D. (the slope of KF) = – (the slope of TV)

3. All the statements below, except one, will prove that GT is perpendicular to TL . Which

statement will NOT prove they are perpendicular?

A. (slope GT ) • (slope TL ) = – 1

B. slope GT = – 1

slope TL

C. slope GT = 1

slope TL

D. slope TL = – 1

slope GT

4. Given: EU GJ and the slope of 2

3EU

Which one statement below is true?

A. The slope of GJ is 2

3

B. The slope of GJ is 3

2

C. The slope of GJ is 2

3

D. The slope of GJ is – 3

2

(cont)

x

y

T

V

K

F

x

y

T

L G

Geometry - 49 -


5. Given: ZU and SV are two distinct lines.

The slope of 1

2ZU , the slope of

3

6SV


A. ZU SV

B. ZU SV

C. ZU and SV are skew lines

D. SV is more steep than ZU

6. Given: (slope of line r ) • (slope of line v ) = – 1 and slope of line r = 4

5


A. (slope of line v ) = 4

5

B. (slope of line v ) = 4

5

C. (slope of line v ) = 5

4

D. (slope of line v ) = 5

4

7. Which statement must be true?

A. (slope k) + (slope t) = 1

B. (slope k) + (slope t) = –1

C. (slope k) • (slope t) = –1

D. (slope k) • (slope t) = 1

8. Which statement below is true?

A. (slope f) is positive

B. (slope f) is the same as the (slope g)

C. (slope f) is greater than (slope g)

D. (slope f) • (slope g) = – 1

9. Given: slope of CK = 3

4, slope of RS = –

3

4, slope of LB =

4

3

Which statement must be true?

A. CK RS

B. CK RS

C. RS LB

D. RS LB

x

y k

t

f

g

x

y

Geometry - 50 -

Unit 3 Review

Note: Diagrams are not drawn to scale.

In problems 1 – 6, identify the pairs of angles below as:

A. Corresponding B. Alternate Interior C. Alternate Exterior

D. Consecutive Interior (Same Side Interior) E. Vertical

1. 3 and 6

2. 2 and 7

3. 4 and 8

4. 5 and 8

5. 3 and 5

6. 1 and 8

In problems 7 - 14 solve for the missing variables. Show all work.

7. 8. 9.

10. 11. 12.

13. 14.

(cont)

1

6

2

8 7

4 3

5

(3x + 11)°

65°

(3x + 15)°

5(x – 9)°

(2x – 18)°

24°

z°

2x° (4y – 4)°

40°

(4y – 10)° 5x°

y°

(3x + 6y)°

4x°

(5x + 6y)°

130°

120°

x°

y° 50°

32°

z° 2x° y°

Geo metry C o urs e Ou tlin e - Weebly

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