Download G CBA 2 Review

Geometry
CBA #2 Review
Name: ______________________________
1. Decide whether the angles are alternate interior angles, same-side
interior angles, corresponding angles, or alternate exterior angles.
a. ∠2 and ∠7
b. ∠5 and ∠4
c. ∠8 and ∠3
d. ∠6 and ∠4
e. ∠1 and ∠5
2. Line e intersects trapezoid ABCD. Sketch a diagram that meets the following conditions.
𝐴𝐵 and 𝐷𝐶 are parallel, ∠1 and ∠6 are alternate exterior angles, ∠2 and ∠3 are same-side
interior angles, and ∠4 and ∠5 are each supplementary to ∠3.
3. Use the figure at the right.
a. Which is a pair of alternate interior angles?
A. ∠3 and ∠6
B. ∠6 and ∠5
C. ∠2 and ∠7
D. ∠4 and ∠6
b. Which angle corresponds to ∠7?
A. ∠1
B. ∠3
C. ∠4
D. ∠6
c. Which pair of angles are alternate exterior angles?
A. ∠1 and ∠5
B. ∠3 and ∠6
C. ∠5 and ∠8
D. ∠1 and ∠8
d. Which pair of angles are same-side interior angles?
A. ∠1 and ∠5
B. ∠3 and ∠6
C. ∠4 and ∠8
D. ∠3 and ∠5
4. Identify all the numbered angles that are congruent to the given angle. Justify your answers.
a.
b.
5. Find m∠1 and m∠2. Justify each answer.
a.
b.
c.
d.
6. Supply the missing reasons in the two-column proof.
Given: a b , c d
Prove: ∠1 and ∠4 are supplementary.
Statements
1) ∠1 ≅ ∠2
2) c d
Reasons
1) a.
2) Given
3) ∠2 and ∠3 are supplementary.
3) b.
4) a b
4) Given
5) ∠3 ≅ ∠4
5) c.
6) ∠1 and ∠4 are supplementary.
6) d.
7. Find the value of x. Then find the measure of each labeled angle.
a.
b.
8. Use the figure at the right.
a. What is the value of x?
A. 10
B. 25
C. 30
D. 120
b. What is the measure of ∠1?
A. 45
B. 60
D. 125
C. 120
9. Which lines or segments are parallel? Justify your answer.
a.
b.
c.
d.
e.
f.
10. Find the value of x for which a b .
a.
b.
c.
d.
11. Use the given information to determine which lines, if any, are
parallel. Justify each conclusion with a theorem or postulate.
a. ∠8 is supplementary to ∠9.
b. ∠7 ≅ ∠4
c. ∠9 is supplementary to ∠12.
d. ∠9 ≅ ∠11
12. Assume a, b, c, and d are distinct lines in the same plane. For each combination
of relationships, tell how a and c relate. Justify your answer.
a. a ⊥ b; b ⊥ c
b. a ⊥ b; b ║ c
13. If 𝒂 ⊥ 𝒃, 𝒃 ⊥ 𝒄, 𝒄 ∥ 𝒅, and 𝒅 ∥ 𝒆, which is not true?
A. 𝑎 ∥ 𝑒
B. 𝑎 ∥ 𝑐
C. 𝑎 ∥ 𝑑
D. 𝑏 ∥ 𝑑
14. Find the value of each variable.
a.
b.
c.
15. Given: AD and BE bisect each other.
AB ≅ DE
Prove: ΔACB ≅ ΔDCE
Statements
1) AD and BE bisect each other.
AB ≅ DE
Reasons
1) Given
2) AC ≅ CD , BC ≅ CE
2) a.
3) ΔACB ≅ ΔDCE
3) b.
16. If ΔABC ≅ ΔJKL, which of the following must be a
correct congruence statement?
A. ∠A ≅ ∠L
B. ∠B ≅ ∠K
C. AB ≅ JL
D. ΔBAC ≅ ΔLKJ
17. Find the values of the variables.
a. ΔXYZ ≅ ΔFED
b. ΔABD ≅ ΔCDB
18. ΔFGH ≅ ΔQRS. Find the measures of the given angles or the lengths of the given sides.
a. m∠F = x + 24; m∠Q = 3x
b. GH = 3x − 2; RS = x + 6
19. The triangles at the right are congruent. Which of the following
statements must be true?
A. ∠A≅∠D
B. ∠B≅∠E
C. 𝐴𝐵 ≅ 𝐷𝐸
D. 𝐵𝐶 ≅ 𝐷𝐹
20. Given the diagram at the right, which of the following must be true?
A. ∆𝑋𝑆𝐹 ≅ ∆𝑋𝑇𝐺
B. ∆𝐹𝑋𝑆 ≅ ∆𝑋𝐺𝑇
C. ∆𝑆𝑋𝐹 ≅ ∆𝐺𝑋𝑇
D. ∆𝐹𝑋𝑆 ≅ ∆𝐺𝑋𝑇
21. If ∆𝑹𝑺𝑻 ≅ ∆𝑿𝒀𝒁, which of the following need not be true?
A. ∠R≅∠X
B. ∠T≅∠Z
C. 𝑅𝑇 ≅ 𝑋𝑍
D. 𝑆𝑅 ≅ 𝑌𝑍
22. If ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭, m∠A = 50, and m∠E = 30, what is m∠C?
A. 30
B. 50
C. 100
D. 120
23. If ABCD ≅ QRST, m∠A = x – 10, and m∠Q = 2x – 30, what is m∠A?
A. 10
B. 20
C. 30
D. 40
24. Would you use SSS or SAS to prove the triangles congruent? If there is not enough
information to prove the triangles congruent by SSS or SAS, write not enough
information. Explain your answer.
a.
b.
c.
d.
e.
f.
g.
h.
25. Reasoning Suppose AB ≅ DE, ∠B ≅ ∠E, and AB ≅ BC. Is ∆ABC congruent to ∆DEF?
Explain.
26. Given: BD is the perpendicular bisector of AC .
Prove: ∆BAD ≅ ∆BCD
Statements
1) BD is the perpendicular bisector of AC .
Reasons
1) Given
2) AD ≅ CD
3) ∠ADB and ∠CDB are right angles
4) a.
5) c.
6) e.
2) Definition of segment bisector
3) Definition of perpendicular
4) b.
5) d.
6) f.
27. Which pair of triangles can be proved congruent by SSS?
28. Which pair of triangles can be proved congruent by SAS?
29. What additional information do you need to prove ∆𝑵𝑶𝑷 ≅ ∆𝑸𝑺𝑹?
A. ∠P≅∠S
B. ∠O≅∠S
C. 𝑃𝑁 ≅ 𝑆𝑄
D. 𝑁𝑂 ≅ 𝑄𝑅
30. What additional information do you need to prove ∆𝑮𝑯𝑰 ≅ ∆𝑫𝑬𝑭?
A. ∠F≅∠G
B. 𝐻𝐼 ≅ 𝐸𝐹
C. 𝐻𝐼 ≅ 𝐸𝐷
D. 𝐺𝐼 ≅ 𝐷𝐹
31. Given: BD is the angle bisector of ∠ABC and ∠ADC.
Prove: ∆ABD ≅ ∆CBD
Statements
1) a.
Reasons
1) b.
2) c.
3) BD ≅ BD
2) Definition of ∠ bisector
3) d.
4) e.
4) ASA
32. Which pair of triangles can be proven
congruent by the ASA Postulate?
33. For the ASA Postulate to apply, which
side of the triangle must be known?
A. the included side
C. the longest side
B. the shortest side
D. the non-included side
34. Which pair of triangles can be proven
congruent by the AAS Theorem?
35. For the AAS Theorem to apply, which side
of the triangle must be known?
A. the included side
B. the shortest side
C. the longest side
D. the non-included side
36. Given: BD ⊥ AB , BD ⊥ DE , BC ≅ DC
Prove: ∠A ≅ ∠E
Statements
Reasons
1) BD ⊥ AB , BD ⊥ DE
1) a.
2) ∠CDE and ∠CBA are right angles.
2) Definition of right angles
3) ∠CDE ≅ ∠CBA
3) b.
4) c.
4) Vertical angles are congruent.
5) BC ≅ DC
5) d.
6) e.
6) f.
7) ∠A ≅ ∠E
7) g.
37. Based on the given information in the figure at the
right, how can you justify that ∆𝑱𝑯𝑮 ≅ ∆𝑯𝑱𝑰?
A. ASA
B. AAS
C. SSS
D. SAS
38. In the figure at the right the following is true: ∠𝑨𝑩𝑫 ≅ ∠𝑪𝑫𝑩
and ∠𝑫𝑩𝑪 ≅ ∠𝑩𝑫𝑨. How can you justify that ∆𝑨𝑩𝑫 ≅ ∆𝑪𝑫𝑩?
A. SAS
B. ASA
C. SSS
D. CPCTC
39. ∆𝑩𝑹𝑴 ≅ ∆𝑲𝒀𝒁. How can you justify that 𝒀𝒁 ≅ 𝑹𝑴?
A. SAS
B. ASA
C. SSS
D. CPCTC
40. Give the expression for the angle measure for ∠𝑫𝑨𝑪.
a.
b.
D
A
A
D
B
x
y
x
C
C
41. Which statement cannot be justified given only that ∆𝑷𝑩𝑱 ≅ ∆𝑻𝑰𝑴?
A. 𝑃𝐵 ≅ 𝑇𝐼
B. ∠BJP ≅ ∠IMT
C. ∠B ≅ ∠I
D. 𝐽𝑃 ≅ 𝑀𝐼
42. In the figure at the right, which theorem or postulate can you
use to prove ∆𝑨𝑫𝑴 ≅ ∆𝒁𝑴𝑫?
A. ASA
B. AAS
C. SSS
D. SAS
43. In the figure at the right, which theorem or postulate can
you use to prove ∆𝑲𝑮𝑪 ≅ ∆𝑭𝑯𝑬?
A. ASA
B. AAS
C. SSS
D. SAS
44. Find the values of x and y.
a.
b.
d.
e.
c.
f.
45. Use the properties of isosceles and equilateral triangles to find the measure of the
indicated angle.
a. m∠ACB
b. m∠DBC
c. m∠ABC
46. Find the values of m and n.
a.
b.
c.
47. Which additional piece of information would allow you to
prove that the triangles are congruent by the HL theorem?
A. 𝐴𝐵 ≅ 𝐷𝐸
B. m∠DFE = 40
C. m∠F = m∠ABC
D. 𝐴𝐶 ≅ 𝐷𝐹
48. For what values of x and y are the triangles shown congruent?
A. x = 1, y = 4
B. x = 4, y = 1
C. x = 2, y = 4
D. x = 1, y = 3
49. Two triangles have two pairs of corresponding sides that are congruent. What else must
be true for the triangles to be congruent by the HL Theorem?
A. The included angles must be right angles.
B. They have one pair of congruent angles.
C. Both triangles must be isosceles.
D. There are right angles adjacent to just one pair of congruent sides.
50. Which of the following statements is true?
A. ∆𝐵𝐴𝐶 ≅ ∆𝐺𝐻𝐼 by SAS.
B. ∆𝐷𝐸𝐹 ≅ ∆𝐺𝐻𝐼 by SAS.
C. ∆𝐵𝐴𝐶 ≅ ∆𝐷𝐸𝐹 by HL.
D. ∆𝐷𝐸𝐹 ≅ ∆𝐺𝐻𝐼 by HL.
51. Are the given triangles (to the far right) congruent by the HL Theorem? Explain.
52. Determine if the given triangles are congruent by the Hypotenuse-Leg Theorem. If so,
write the triangle congruence statement.
a.
b.
c.
d.
53. Find the value of x in the diagram to the right.
x
25°
54. What is the common angle of ∆𝑷𝑸𝑻 and ∆𝑹𝑺𝑸?
A. ∠PQT
B. ∠SRQ
C. ∠SPT
D. ∠SUT
55. Use the figure to the right and the information below
Given: ∆𝒁𝑾𝑿 ≅ ∆𝒀𝑿𝑾, 𝒁𝑾 ∥ 𝒀𝑿
Prove: ∆𝒁𝑾𝑹 ≅ ∆𝒀𝑹𝑿
a. Which corresponding parts statement is needed to prove ∆𝑍𝑊𝑅 ≅ ∆𝑌𝑅𝑋?
A. 𝑍𝑊 ≅ 𝑌𝑋
B. ∠ZWR ≅ ∠YXR
C. ∠Z ≅ ∠R
D. 𝑊𝑋 ≅ 𝑊𝑋
b. A classmate writes the statement ∠ZRW ≅ ∠YRX to help prove the congruence of the
triangles. What reason should the classmate give?
A. Given
B. Angles cut by a bisector are congruent.
C. Base angles of an isosceles triangle are congruent.
D. Vertical angles are congruent.
c. After using the congruence statements from parts a and b, which statement can be used to
prove the triangles congruent?
A. 𝑊𝑋 ≅ 𝑊𝑋
B. ∠ZWR ≅ ∠RYX
C. ∠Z ≅ ∠Y
D. 𝑊𝑅 ≅ 𝑅𝑋
d. Which theorem or postulate will prove ∆𝑍𝑊𝑅 ≅ ∆𝑌𝑅𝑋?
A. ASA
B. AAS
C. SSS
D. SAS
56. Find the value of x.
a.
b.
57. Use the figure at the right.
a. What is the relationship between 𝐿𝑁 and 𝑀𝑂?
b. What is the value of x?
c. Find LM.
d. Find LO.
58. Use the figure at the right.
a. According to the figure, how far is A from 𝐶𝐷? From 𝐶𝐵?
b. How is 𝐶𝐴 related to ∠DCB? Explain.
c. Find the value of x.
d. Find m∠ACD and m∠ACB.
e. Find m∠DAC and m∠BAC.
c.
59. Find the indicated values of the variables and measures.
a. x, BA, DA
b. r, UW
60. Use the figure at the right.
a. Which ray is a bisector of /ABC?
A. 𝐵𝐶
B. 𝐵𝐴
C. 𝐵𝐷
b. What is GH?
A. 5
B. 10
C. 15
c. What is the value of y?
A. 2
B. 4
C. 16
d. What is m∠DBE?
A. 20
B. 30
C. 40
e. What is m∠ABE?
A. 20
B. 30
C. 40
f. If m∠FBA = 7x + 6y, what is m∠FBA?
A. 40
B. 44
C. 47
61. Find the value of x.
a.
b.
c. m, p
D. 𝐵𝐹
D. 25
D. 20
D. 50
D. 60
D. 60
c.
62. Name the point of concurrency of the angle bisectors.
a.
b.
63. Use the figure at the right.
a. Which point is the incenter of ∆𝐴𝐵𝐶?
A. X
B. T
C. R
D. Y
b. Which point is the circumcenter?
A. X
B. T
C. R
D. Y
c. Which segment is an angle bisector?
A. 𝐵𝑋
B. 𝑆𝑋
C. 𝐴𝑆
D. 𝑅𝑍
d. Which segment is a perpendicular bisector of ∆𝐴𝐵𝐶?
A. 𝐵𝑋
B. 𝑆𝐴
C. 𝐴𝑆
D. 𝑅𝑍
e. If RC = x + 3 and RA = 3x – 3, what is the value of x?
A. 3
B. 6
C. 7
D. 9
d.
64. Is 𝑨𝑩 a median, an altitude, or neither? Explain
a.
b.
65. Name the centroid.
67.
a.
b.
c.
d.
c.
d.
66. Name the orthocenter.
Name each segment.
A median in ∆STU
An altitude in ∆STU
A median in ∆SBU
An altitude in ∆CBU
68. Z is the centroid of ∆𝑨𝑩𝑪. If AZ = 12, what is ZY?
A. 6
B. 9
C. 12
D. 18
69. What is the best description of 𝑨𝑩?
A. altitude
B. perpendicular bisector
C. median
D. angle bisector
70. What is the best description of P?
A. incenter
B. centroid
C. circumcenter
D. orthocenter
71. Use the figure to the right.
a. Which is an altitude of ∆𝑿𝒀𝒁?
A. 𝐴𝑍
B. 𝑋𝐵
C. 𝑋𝑌
D. 𝑍𝑌
b. Which is a median of ∆𝑿𝒀𝒁?
A. 𝐴𝑍
B. 𝐵𝑋
C. 𝑋𝑌
D. 𝑍𝑌
72. M is the centroid of ∆𝑸𝑹𝑺, and QM = 22x + 10y. What expressions can
you write for MV and QV?
73. For each figure, list the angles of each triangle in order from smallest to largest.
a.
b.
c.
d.
74.For each figure (or description), list the sides of each triangle in order from shortest to longest.
a. ∆𝐴𝐵𝐶, with m∠A = 99,
b.
c.
m∠B = 44, and m∠C = 37
d. ∆𝑀𝑁𝑂, with m∠A = 122,
m∠B = 22, and m∠C = 36
75. Determine which side is shortest in the diagram.
a.
b.
76. Can a triangle have sides with the given lengths? Explain.
a. 8 cm, 7 cm, 9 cm
b. 7 ft, 13 ft, 6 ft
c. 20 in., 18 in., 16 in.
d. 3 m, 11 m, 7 m
77. The lengths of two sides of a triangle are given. Write a compound inequality for the
possible lengths for the third side.
a. 5, 11
b. 12, 12
c. 25, 10
d. 6, 8
78. Which of the following could be lengths of sides of a triangle?
A. 11, 15, 27
B. 13, 14, 32
C. 16, 19, 34
D. 33, 22, 55
79. ∆𝑨𝑩𝑪 has the following angle measures: m∠A = 120, m∠B = 40, and m∠C = 20.
Which lists the sides in order from shortest to longest?
A. 𝐶𝐵, 𝐵𝐴, 𝐴𝐶
B. 𝐵𝐴, 𝐴𝐶, 𝐶𝐵
C. 𝐴𝐶, 𝐵𝐴, 𝐶𝐵
D. 𝐶𝐵, 𝐴𝐶, 𝐵𝐴
80. ∆𝑹𝑺𝑻 has the following side lengths: RS = 7, ST= 13, and RT = 19. Which lists the
angles in order from smallest to largest?
A. ∠R, ∠S, ∠T
B. ∠S, ∠T, ∠R
C. ∠T, ∠S, ∠R
D. ∠T, ∠R, ∠S
81. A triangle has side lengths 21 and 17. Which is a possible length for the third side?
A. 2
B. 4
C. 25
D. 39
82. Look at ∆𝑳𝑴𝑵. Which lists the angles in order from
the smallest to the largest?
A. ∠L, ∠M, ∠N
B. ∠M, ∠N, ∠L
C. ∠N, ∠M, ∠L
D. ∠M, ∠L, ∠N
83. What are the possible lengths for x, the third side of a triangle, if two sides are 13 and 7?
A. 6 < x < 20
B. 7 < x < 13
C. 6 < x < 20
D. 7 < x < 13