Int. Geometry Final Exam Review 1

A CB

D E

12

3

4 5 6

A B

C

Things to note about this review and the Final Exam: 1. This review packet covers major ideas from the year, but it does not show

examples of all types of problems. 2. Tests and quizzes are another excellent place to study. 3. The test will be a mixture of multiple choice, free response, and proofs. Multiple Choice: Select the best answer. 1. If B is the midpoint of 𝐴𝐶, 𝐴𝐵 = 4𝑥 + 12, and 𝐵𝐶 = 7𝑥 − 21, then what is AC?

(U1L2,3) 2. In the diagram ∠𝐴𝐵𝐷 ≅ ∠𝐵𝐸𝐶. If 𝑚∠𝐴𝐵𝐷 = 2𝑥° and 𝑚∠𝐷𝐵𝐸 = 5𝑥°, then

what is 𝑚∠𝐷𝐵𝐶? (U1L4) 3. What do the following symbols represent? (U1L5,7,9) a) ~ b) ↔ c) → d) ∴ e) ∧ f) ∨ 4. If p = true, q = false, and r = true, then what is the truth value of: ~𝑞 ∧ 𝑝 ∨ ~𝑟 (U1L5) 5. In the Venn diagram below Sets A, B, and C relationships are shown. Each

individual region is labeled with a number. List the numbers that each description is describing: (U1L6)

a) in A and in B b) in A or in C c) not in B and in A d) not in A and not in B and not in C

Int. Geometry Final Exam Review 2

5x+20( )°7x-10( )°

A

B

C D

6. Write a conditional that is logically equivalent to “If it rains, the grass grows.” (U1L8)

7. Which condional’s inverse is true? (U1L7,8)

(a) If two lines intersect to form a right angle, then they are perpendicular. (b) If m W∠ is less than 45°, then m W∠ is acute. (c) If point K is the midpoint of 𝐽𝐿, then points J, K, and L are collinear. (d) If two rays are opposite rays, then they have a common endpoint. Questions 8-9: Determine if the argument pattern is valid and state the pattern name. (U1L9) 8. If the butler committed the crime, then the maid didn’t do it. The maid didn’t do it. Therefore, the butler committed the crime. 9. 𝑏 → 𝑎 𝑐 → 𝑏 ∴ 𝑐 → 𝑎 Questions 10-11: Given the statements, write a conclusion to create a valid argument. If it is not possible then write none. (U1L9) 10. If you pass this test, then you will get an A for the course. You do not pass the test. 11. All Senators are over 30 years old. (hint: rewrite this in if-then form) Jose is 25 years old. 12. The supplement of an angle is 12 more than 3 times the complement of the angle.

What is the measure of the angle? (U2L5) 13. Determine if 𝐴𝐵 ⊥ 𝐶𝐷 (U2L5,6)

Int. Geometry Final Exam Review 3

l

m

x°

130°

60°

u°w°

t°

s°

r°

x°

P

C

DA

B

EF

84°

6x( )°5y+4( )°

3y+10( )°

2x-5( )°y°

x°

14. Suppose you want to try an indirect proof for the following: “If two coplanar lines are perpendicular to the same line, then the two lines are parallel.” What will the first line of the proof need to be? (U2L4)

15. Line l is parallel to line m. What is the value of x? (U3L2,3)

16. AB , CD, and EF intersect at P. If r = 90, s = 50, t = 60, u = 45, and w = 50,

what is the value of x? (U3L5, U2L5)

17. Solve for the variables to make 18. Solve for the variables. the lines parallel (U3L2 & U3L3) (U3L2 & U3L3)

Int. Geometry Final Exam Review 4

x°5y°

3y°

7y°

l

m35°

x°78°B

A

D

A

B

CD

A

B

C

D

A

B

C

G

F

H

I

E

4x-8

1614

12

B

A C

E

F D

19. Find the value of x. 20. In the figure, for which (U3L5 & U3L6) value of x is 𝑙 ∥ 𝑚? (U3L4 & U3L5)

Questions 21-24: In each of the diagrams below, name all pairs of congruent triangles you can identify (without drawing more segments or naming more points). (a) Write down the triangle congruence statements and (b) explain why the triangles are congruent (SSS, SAS, etc). If there are not any pairs of triangles in a given diagram that are congruent, state none. (U4L2-5 & U4L7) 21. 22. (a)_______________ (a)_______________ (b)_______________ (b)_______________ 23. 24. (a)_______________ (a)_______________ (b)_______________ (b)_______________ 25. It is known that ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐹𝐷. What is the value of x? (U4L1)

Int. Geometry Final Exam Review 5

134°

x°7x-8

3x+85x+6

80°50°D

E

F

43

21

A

B

CD

81°

40°

60°

61°

D

A

B

C

26. Find the value of x. 27. What is the perimeter of (U4L6 & U3L2 & U3L5-6) the triangle? (U4L6) Questions 28-30: Determine if the statement is true or false from the diagram. (U5L1) 28. 𝑚∠3 > 𝑚∠2 29. ∠𝐴 + ∠1 = ∠4 30. 𝐴𝐶 > 𝐴𝐷 31. The diagram is not drawn to scale. If each diagram were drawn accurately, which

segment shown would be the shortest? (U5L2) 32. In triangle ABC, it is known that AB > BC and AB < AC. Order the angles from

least to greatest measure. (U5L2) 33. Two sides of a triangle are 22 and 74. What is the possible range of values of the

3rd side to form a triangle? (U5L3) 34. If the lengths of two sides of an isosceles triangle are 7 and 15, what is the

perimeter of the triangle? (U5L3)

Int. Geometry Final Exam Review 6

3y-30( )°

5x-8

2x+160° 30°

20

1810

A

B

C D

E

F

22

64

12

3

C

A

D

B

E125°

55°H

F

I

G J

15

9

12

A

D

B

C

241216

8

E

G

F

H I

J

35. What is the difference between an altitude and a median of a triangle? (U5L4) 36. a) Find the value of x and y given that ∆𝐴𝐵𝐶~∆𝐸𝐹𝐷. (U5L5) b) What is the scale factor? (U5L5) c) Hence, or otherwise, what is BC? (U5L5) Directions 37-40: Tell whether the triangles are similar or not similar. If they are similar state, the postulate or theorem used and write a similarity statement. (U5L6-7) 37. 38. 39. 40.

Int. Geometry Final Exam Review 7

4

x9

2x

D

CB

A

y

x

9

4

18 12

C

A

B

D

E

7

2

5x

12x

x+8

41. Solve for x. 42. Find the values of x and y. (U5L8) (U5L8 & U5L6 & U5L1)

43. In right triangle ABCΔ , DE BC⊥ . If AB = 6, AC = 8, BC = 10, and DE = 4, find

EC. (U5L6 & U5L1) Questions 44-46: Classify the triangle as acute, obtuse, right. If a triangle cannot be formed, then write not a triangle. (U6L3 & U5L3) 44. 14, 48, 50 45. 6, 2, 4 46. 8, 7, 6 47. Solve for x in simplest 48. Solve for x in simplest

radical form. (U6L1 & U6L2) radical form. (U6L1 & U6L2)

Int. Geometry Final Exam Review 8

10

8

30°

DCB

A

E

60°

C

A

D

B

7

4 2

45°A C

B

49. In the figure, what is the length of AB ?(U6L4,5)

50. If BC = DC and AD = AB, what is the value of ABCD

? (U6L4,5 U4L6)

51. What is the length of side BC? (hint create a special right triangle) (U6L5) 52. Is it possible for a polygon to have an interior angle sum of 4160o? (U7L2) 53. How do you find one interior angle of a regular polygon? One exterior angle? (U7L2)

Int. Geometry Final Exam Review 9

2y-3( )°47°

4x-8

23

A

B C

D

x6

y°74°10

A

B C

D

y°x

32°

A B

CD

x°

54. A regular pentagon and a regular octagon share a common side. A side of each figure is extended to form a quadrilateral. What is the value of x? (U7L2)

55. Given that quadrilateral ABCD is a parallelogram find the values of x and y. (U7L3) 56. What are the ways to show a quadrilateral is a parallelogram? (U7L4) 57. Given rhombus ABCD, find the values of x and y. (U7L5) 58. Given rectangle ABCD and AC = 10, find the values of x and y. (U7L5)

Int. Geometry Final Exam Review 10

20

12

15

15

D

A

B

C

87 6 5

4321

S

N

P O

M

87 6 5

4321

S

N

P O

M

87 6 5

4321

S

N

P O

M

87 6 5

4321

S

N

P O

M

59. In isosceles trapezoid ABCD, 𝐴𝐵 ∥ 𝐶𝐷, m A x°∠ = and ( )2 45m B x °∠ = − . Find m C∠ and m D∠ . (U7L8)

60. The lengths of the bases of a trapezoid are 8 and 22. What is the length of the

median of the trapezoid? (U7L8) 61. Given kite ABCD, find AC, BD, and BC. (U7L9) Directions 62-65: (a) Give the most accurate name for the quadrilateral

(b) Write the reason why – you may write a phrase or the entire theorem or definition that justifies your answer.

(U7L10) 62. 𝑃𝑂 ∥ 𝑀𝑁; PO = MN 63. ; MO NP PS SN⊥ =

64. ; m PON m PMN m OPM m ONM∠ = ∠ ∠ = ∠ 65. 1 5; m m PN OM∠ = ∠ =

Int. Geometry Final Exam Review 11

1

3

C

B D

42°

17b

a

66. Find the indicated trigonometric ratios. Write your answers in simplest radical form. (U8L2,3,5)

sinD = cscD = cosD = secD = tanD = cotD =

67. One acute angle of a right triangle has measure no. If 3sin5

n = , what is the value

of tan n ? (U8L2,3) 68. What is the value of a to the nearest tenth? (U8L2,3) 69. What is the angle of elevation of the sun when a tree 18.2 meters tall casts a

shadow 27.9 meters long? Round to the nearest degree. (U8L6,7) 70. In ∆𝑋𝑌𝑍, 𝑚∠𝑍 = 90Q and sin X = VW

VX . If 𝑋𝑌 = 120, then what is XZ?

Round to the nearest tenth. (U8L2,3)

Int. Geometry Final Exam Review 12

Y

X

O

A B

A

O

C

B

71. The sides of a rectangle are 15 and 30. What is the measure of the obtuse angle formed by the diagonals? Round to the nearest tenth. (U8L6)

72. The angle of depression from the top of a tower to a point A is 23 . The distance

from A to the base, B, of the tower is 80 m. How tall is the tower to the nearest meter? (U8L6,7)

73. If the radius of circle O is 17 and AB = 30, what is the length of segment XY?

(U9L3,4) 74. Triangle ABC is inscribed in the circle O and diameter AC . If AB = AO, what is

the degree measure of 𝐴𝐵? (The picture may not be drawn to scale). (U9L2)

Int. Geometry Final Exam Review 13

9

135Z

Y

X

S

T

R

A

DC

B

XY

O

P

816

x

A

75. R, S, and T are all points of tangency. What is the perimeter of XYZΔ ? (U9L7,8) 76. The measure of 𝐵𝐶𝐷 is 152 . What is the measure of C∠ ? (U9L6) 77. XY is tangent to both circle O and circle P. Given that OP = 40, and the radii of

circles O and P are 31 and 7, respectively, find XY. (U9L7,8) 78. A tangent is drawn to circle A. What is the value of x? (U9L7,8 )

Int. Geometry Final Exam Review 14

135°

Y

X

W Z

6x-50( )°2x+30( )°

65°

A

y°

135°

5x( )°

67°A

8x-30( )°2x+16( )° 34°

A

610

x8

x

6

18

Questions 79-81: Find the value of the variables in circle A. Lines that appear to be tangents are tangents. 79. (U9L10) 80. (U9L5,6,2) 81. (U9L11) Directions 82-83: Chords, secants, and tangents are shown. Find the value of x. 82. (U9L11) 83. (U9L12) 84. If WZ and XY are diameters with lengths 12, what is the area of the shaded

region? (U10L4)

Int. Geometry Final Exam Review 15

w

P

A C

B

D

85. In the figure, point P is the center of each circle. The circumference of the larger circle exceeds the circumference of the smaller circle by 12π . What is the width, w, of the region between the two circles? (U10L1,2)

86. Two similar pyramids have volumes 54 and 250. If the smaller has lateral area

18, what is the lateral area of the larger? (U10L10) 87. A cone has radius 5 and height 12. A cylinder with radius 10 has the same

volume as the cone. What is the cylinder’s height? (U10L7,8) 88. The water level of a swimming pool, 75 feet by 42 feet, is to be raised 4 inches.

How many gallons of water must be added to accomplish this? (1 cubic foot = 7.48 gallons) (U10L7)

89. The ratio of AD to DC is 3 to 2. If the area of ABCΔ is 40, what is the area of

BDCΔ ? (U10L5,6)

Int. Geometry Final Exam Review 16

II

I

13

9

8

12

10

E

F

B

C D

A

90. Find the ratio of areas for region I to region II. (U10L5,6) 91. A cone has its height multiplied by 2 and its radius multiplied by 3. What effect

does this have on the volume of the cone? The volume of the prism is multiplied by ________ (U10L8)

92. The volume of a cone is 400 3cm and the radius of the base is 5 cm. Find its

altitude. (U10L8) 93. Find the surface area and volume of the figure below. (U10L7,9)

94. Find the length of CF. AB = 12, AD = 10, CE = 8. (U10L1)

Int. Geometry Final Exam Review 17

95. Find the (a) midpoint, (b) distance, and (c) slope between the points A(4, -5) and B(-6, 2). (U11L1)

96. Tell the center and radius of the equation ( )2 24 ( 3) 17x y+ + − = (U11L2,3) 97. Write the circle equation given the center (-3, 1) and radius 5. (U11L2,3) 98. The center of a circle is (-20, 15) and the point (-8, 0) is on the circle. Write the

equation of the circle. (U11L2,3)

99. What is an equation of the line through (-4, 7) and perpendicular to 2 53

y x= + ?

(U11L4)

100. Points (2, 2) and (8, v) lie on a line with slope 12

. What is the value of v.

(U11L4)

Int. Geometry Final Exam Review 18

B (?, ?)

O A (2b, 0)

C(9,0)O(0,0)

A(??, 6) B(15, ??)

101. The midpoint of AB is (3, 4). If the coordinates of B are (6, 6), what are the coordinates of A? (U11L1)

102. The points (3, 1) and (5, y) are 13 units apart. What does y equal? (U11L1) 103. Fill in the missing coordinate values of the given figure: (U11L4) a) Equilateral Triangle b) Parallelogram 104. Whatarethecoordinatesof ABCΔ afterareflectionintheline y x= ? (U11L6) A’(,)

B’(,)

C’(,)

Int. Geometry Final Exam Review 19

105. Whatarethecoordinatesoftheimageof ABCΔ aftera90°rotationabouttheorigin?(U11L7)

A’(,)

B’(,)

C’(,)

106. Whatisthepre-image,P,ofP’(11,-4)usingthetransformation

(x,y)→(3x,4y)?(U11L5) 107. Howmanylinesofsymmetrydoesthefollowingfigurehaveandtellthe

rotationalsymmetry.(U11L9)a) b)

Int. Geometry Final Exam Review 20

B

A C

B

A C

B

A C

B

A C

Constructions: Create the indicated construction (U12) 108. a) The altitude to AC b) line parallel to AC through B 109. a) midpoint of AC b) incenter of the triangle

Int. Geometry Final Exam Review 21

ED

FB C

A

Z

W

Y

X

2

1

E

F

BA

D C

c

d

a b

1211109

87654321

21D

A

B

C

Proofs: 110. Given: AC = AB; ; DF AB EF AC⊥ ⊥ Prove: BD EF CE DF⋅ = ⋅ 111. Given: 𝑎 ∥ 𝑏 and 𝑚∠2 ≠ 𝑚∠10 Prove: is not parallel to c d 112. Given: WX = YZ; ; WZ ZY WX XY⊥ ⊥ Prove: 𝑊𝑍 ∥ 𝑋𝑌 113. Given: 𝐴𝐵 ∥ 𝐶𝐷, 𝐴𝐵 ≅ 𝐶𝐷, ∠1 ≅ ∠2 Prove: DE BF≅ 114. Given: 𝑚∠𝐶 > 𝑚∠2, and 𝐴𝐵 ∥ 𝐶𝐷 Prove: 𝐵𝐷 > 𝐶𝐵

Int. Geometry Final Exam Review 22

V

XW

Y

Z

A

B

C

D

E C

A B

D

115. Given: Diagram as shown. Prove: WX YV XV ZY⋅ = ⋅ 116. Given: AB = AD and BC > CD Prove: 𝑚∠𝐵𝐴𝐶 ≠ 𝑚∠𝐷𝐴𝐶 117. Given: Rectangle ABCD and parallelogram ABDE Prove: EACΔ is isosceles Selected Answers: 1. AC = 112 2. 𝑚∠𝐷𝐵𝐶 = 140° 3. a) not (negation) b) if and only iff (biconditional) c) if then (conditional) d) therefore e) and (conjunction) f) or (disjunction) 4. True 5. a) 5 b) 2, 3, 4, 5 c) 3, 4 d) 1

Int. Geometry Final Exam Review 23

6. If the grass doesn’t grow, then it didn’t rain. (the contrapositive) 7. (a) 8. Invalid; Converse Error 9. Valid; Law of Syllogism 10. No conclusion (Inverse error) 11. Jose is not a Senator (law of the contrapositive inference) 12. 51o 13. No (x = 15) 14. Temporally assume that the lines are not coplanar 15. x = 110 16. x = 55 17. x = 16; y = 16 18. x = 38; y = 33 19. x = 60 ( y = 15) 20. x = 47 21. a) ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴 22. a) ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶 b) HL≅ b) SSS≅ 23. a) none 24. a) none b) none (SSA) b) none (AAA) 25. x = 6 26. x = 56.5 27. perimeter = 66 28. True (ext angle inequality thm) 29. False (notation) 30. True (a prop of inequality) 31. 𝐴𝐵 32. ∠𝐴, ∠𝐶, ∠𝐵 33. between 52 and 96 (but not =) 34. 37 ( 7, 7, 15 is not a triangle) 35. median goes to midpoint; altitude forms right angle 36. a) x= 7; y = 40 b) 3:2 c) BC = 30 37. ∆𝐴𝐵𝐸~∆𝐴𝐷𝐶; SSS~ 38. ∆𝐹𝐺𝐽~∆𝐹𝐼𝐻; AA~ 39. none similar 40. ∆𝐸𝐺𝐹~∆𝐻𝐽𝐼; SAS~ 41. 𝑥 = 3 2 42. x = 6; y = 12 43. 𝑥 = XV

_ 44. Right triangle

45. not a triangle 46. Acute triangle 47. 𝑥 = _ `

` 48. x =5

49. 𝐴𝐵 = 2 3 50. 2 51. BC = 5 52. No, it would have 25 X

a sides.

53. int = XbW(def)d

; ext = _VWd

54. 126o

55. 𝑥 = _Xh

; y = 68 56. Lots of ways, see your notes 57. x = 8; y = 74 58. x = 5; y = 64 59. 𝑚∠𝐶 = 𝑚∠𝐷 = 135° 60. Median = 15 61. 𝐴𝐶 = 29; 𝐵𝐷 = 24; 𝐵𝐶 = 4 34 62. Parallelogram 63. Kite 64. Parallelogram

Int. Geometry Final Exam Review 24

B

A C B

A C

65. Isosceles Trapezoid 66. sin𝐷 = _

f;cos𝐷 = X

f;tan𝐷 = 3 ;csc𝐷 = f _

_;sec𝐷 = 2;cot 𝐷 = _

_

67. tan 𝑛 = h

` 68. a = 15.3

69. 33.1o 70. XZ = 21.6 71. 126.6o 72. 34m 73. XY = 9 74. 𝐴𝐵 = 60° 75. 54 76. 𝑚∠𝐶 = 104° 77. XY = 32 78. x = 12 79. x = 31.25 80. x = 15.8; y = 158 81. x = 19 82. x = 4.8 83. x = 12 84. Shaded = 18 un2 85. w = 6 86. 50 un2 87. h = 1 88. 7854 gallons 89. 16 un2 90. 81:403 91. 18 92. Altitude = 48 93. SA=156𝜋; V=288𝜋 94. CF = 9.6 95. a)(-1,3/2)b) 149 ≈ 12.2c)-7/1096. center (-4, 3) radius 17 97. ( )2 23 ( 1) 25x y+ + − =

98. ( )2 220 ( 15) 369x y+ + − = 99. 𝑦 = − _f𝑥 + 1

100. v = 5 101. (0,2) 102. y = 4 or y = -1 103. a) O(0,0); B(b,𝑏 3) b) A(6,6); B(15,6) 104. A’(-3,-1); B’(-4,2); C’(0,3) 105. A’(-3,1); B’(-4,3); C’(-1,2) 106. P XX

_, −1 107. a) Lines = 0; Rotational = 180o

b) Lines = 6; Rotational = 60o You may have a slightly different construction and still be okay.

108. a) b)

Int. Geometry Final Exam Review 25

I

B

A C

Midpoint

B

A C

109. a) b) construct 2 angle bisectors

Note with the proofs, there are multiple solutions to these problems. 110.

111. Temp. assume that 𝑐 ∥ 𝑑. We are given 𝑎 ∥ 𝑏, and when lines are parallel

corresponding angles are congruent meaning 𝑚∠2 = 𝑚∠4 and 𝑚∠4 = 𝑚∠10. This means by transitivity 𝑚∠2 = 𝑚∠10 but this contradicts the given information that 𝑚∠2 ≠𝑚∠10 therefore c is not parallel to d.

112.

Statement Reason 1. AC=AB 1. Given 2. 𝑚∠𝐵 = 𝑚∠𝐶 2. Isosceles Triangle Theorem 3. 𝐴𝐵 ⊥ 𝐷𝐹; 𝐴𝐶 ⊥ 𝐸𝐹 3. Given 4. ∠𝐵𝐷𝐹 and ∠𝐶𝐸𝐹 are right angles 4. Definition of perpendicular lines 5. ∠𝐵𝐷𝐹 ≅ ∠𝐶𝐸𝐹 5. All right angles are congruent 6. ∆𝐵𝐷𝐹~∆𝐶𝐸𝐹 6. AA similarity 7. xy

z{= y|

{| 7. Definition of similar triangles

8. 𝐵𝐷 ∙ 𝐸𝐹 = 𝐶𝐸 ∙ 𝐷𝐹 8. Means-Extreme property (or cross multiplication.

Statement Reason

1. WX=YZ; 𝑊𝑍 ⊥ 𝑍𝑌; 𝑊𝑋 ⊥ 𝑋𝑌 1. Given 2. ∠𝑍 and ∠𝑋 are right angles 2. Definition of perpendicular lines 3. WY = WY 3. Reflexive PoE 4. ∆𝑊𝑍𝑌 ≅ ∆𝑌𝑋𝑊 4. HL≅ 5. ∠𝑍𝑊𝑌 ≅ ∠𝑋𝑌𝑊 5. Definition of Congruent Triangles 6. 𝑊𝑍 ∥ 𝑋𝑌 6. Alternate Interior Angle Theorem

Int. Geometry Final Exam Review 26

113. 114. 115. 116. Temp. assume that 𝑚∠𝐵𝐴𝐶 = 𝑚∠𝐷𝐴𝐶. We know AB=AD and by the reflexive prop.

AC=AC which means that ∆𝐵𝐴𝐶 ≅ ∆𝐷𝐴𝐶 by SAS≅. This means by def. of congruent triangles BC=DC, but this contradicts the fact that BC>CD, therefore our assumption is false and 𝑚∠𝐵𝐴𝐶 ≠ 𝑚∠𝐷𝐴𝐶.

117.

Statement Reason

1. 𝐴𝐵 ∥ 𝐶𝐷; 𝐴𝐵 = 𝐶𝐷; ∠1 ≅ ∠2 1. Given 2. quad. ABCD is a parallelogram 2. A quad. with one pair of parallel and

congruent sides is a parallelogram 3. 𝐴𝐷 ∥ 𝐵𝐶; 𝐴𝐷 = 𝐵𝐶 3. Opposite sides of a parallelogram are

parallel and congruent 4. ∠𝐷𝐴𝐸 ≅ ∠𝐵𝐶𝐹 4. Alternate Interior Angle Theorem 5. ∠𝐵𝐷𝐹 ≅ ∠𝐶𝐸𝐹 5. All right angles are congruent 6. ∆𝐷𝐴𝐸 ≅ ∆𝐵𝐶𝐹 6. ASA≅ 7. 𝐷𝐸 ≅ 𝐵𝐹 7. Definition of Congruent Triangles

Statement Reason

1. 𝐴𝐵 ∥ 𝐶𝐷; 𝑚∠𝐶 > 𝑚∠2 1. Given 2. 𝑚∠1 = 𝑚∠2 2. Alternate Interior Angle Theorem 3. 𝑚∠𝐶 > 𝑚∠1 3. Substitution PoE 4. BD > CB 4. In a triangle the longer side is opposite the

larger angle.

Statement Reason 1. ∠𝑋 ≅ ∠𝑌; ∠𝑊 ≅ ∠𝑍 1. Inscribed angle intercepting the same arc

are congruent 2. ∆𝑋𝑊𝑉~∆𝑌𝑍𝑉 2. AA similarity 3. ��

��= ��

�� 3. Definition of similar triangles

4. 𝑊𝑋 ∙ 𝑌𝑉 = 𝑋𝑉 ∙ 𝑍𝑌 4. Means-Extreme property (or cross multiplication.

Statement Reason 1. Rectangle ABCD and parallelogram ABDE

1. Given

2. AC = BD 2. Diagonals of a rectangle are congruent 3. 𝐴𝐸 = 𝐵𝐷 3. Opposite sides of a parallelogram are congruent 4. 𝐴𝐶 = 𝐴𝐸 4. Transitive PoE 5. ∆𝐸𝐴𝐶 is an isosceles triangle

5. Definition of isosceles triangle