Chapter 5 Triangles and Congruence Name_______________________________

Geometry

Determine m∠1.

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16. Find the lettered angles, a – f, in the picture below. Note that the two lines are parallel.

17. Fill in the blanks in the proof below. Given: The triangle below with interior and exterior

angles. Prove: m∠4 + m∠5 + m∠6 = 360°

Statement Reason

1. Triangle with interior and exterior

angles

Given

2. m∠1 + m∠2 + m∠3 = 180°

3. ∠3 and ∠4, ∠2 and ∠5, and ∠1 and ∠6 are a linear pair

4. Definition of Linear Pair

5. m∠1 + m∠6 + m∠2 + m∠5 + m∠3 + m∠6 = 540°

6. m∠4 + m∠5 + m∠6 = 360°

18. Write a two column proof. Given: ΔABC with right angle B. Prove: ∠A and ∠C are

complementary.

For problems 19 – 26, solve for x.

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27. If ΔRAT ΔUGH, what is also congruent?

28. If ΔBIG ΔTOP, what is also congruent?

For problems 29 – 33, use the picture below.

29. What theorem tells us that ∠FGH ∠FGI?

30. What is m∠FGI and m∠FGH? How do you know?

31. What property tells us that the third side of each triangle is congruent?

32. How does 𝐹𝐺̅̅ ̅̅ relate to ∠IFH?

33. Write the congruence statement for these two triangles.

For problems 34-38, use the picture below.

34. If 𝐴𝐵̅̅ ̅̅ ∥ 𝐷𝐸̅̅ ̅̅ , what angles are congruent? How do you know?

35. Why is ∠ACB ∠ECD? It is not the same reason as #34.

36. Are the two triangles congruent with the information you currently have? Why or why

not?

37. If you are told that C is the midpoint of 𝐴𝐸̅̅ ̅̅ and 𝐵𝐷̅̅ ̅̅ , what segments are congruent?

38. Write a congruence statement for the two triangles.

For problems 39-42, determine if the triangles are congruent. If they are, write the congruence

statement.

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43. Suppose the two triangles below are congruent. Write a congruence statement for these

triangles.

44. Explain how we know that if the two triangles in #43 are congruent, then ∠B ∠Z.

For problems 45-48, determine the measure of the angles in each triangle.

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49. Fill in the blanks in the Third Angle Theorem proof below. Given: ∠A ∠D, ∠B ∠E.

Prove: ∠C ∠F

For problems 50-54, determine if the Reflexive, Symmetric, or Transitive Properties of

Congruence is used.

50. ∠A ∠B and ∠B ∠C, then ∠A ∠C

51. 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐵̅̅ ̅̅

52. ΔXYZ ΔLMN and ΔLMN ΔXYZ

53. ΔABC ΔBAC

54. What type of triangle is ΔABC in #53? How do you know?

Use the following diagram in problems 55 and 56.

55. Mark the diagram with the following information. 𝑆𝑇̅̅̅̅ ∥ 𝑅𝐴̅̅ ̅̅ , 𝑆𝑅̅̅̅̅ ∥ 𝑇𝐴̅̅ ̅̅ , 𝑆𝑇̅̅̅̅ ⊥ 𝑇𝐴̅̅ ̅̅ and 𝑆𝑅̅̅̅̅ ,

𝑆𝐴̅̅̅̅ and 𝑅𝑇̅̅ ̅̅ perpendicularly bisect each other.

56. Using the given information and your markings, name all of the congruent triangles in the

diagram.

For problems 57-64, are the pairs of triangles congruent? If so, write the congruence statement.

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For problems 65-70, state the additional piece of information needed to show that each pair of

triangles are congruent.

65. Use SAS

66. Use SSS

67. Use SAS

68. Use SAS

69. Use SSS

70. Use SAS

For problems 71-73, fill in the blanks in the proofs below.

71. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ , 𝐵𝐸̅̅ ̅̅ ≅ 𝐶𝐸̅̅̅̅ Prove: ΔABE ΔACE

72. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ , 𝐴𝐶̅̅ ̅̅ ≅ 𝐷𝐵̅̅ ̅̅ Prove: ΔABC ΔDCB

73. Given: B is a midpoint of 𝐷𝐶̅̅ ̅̅ and 𝐴𝐵̅̅ ̅̅ ⊥ 𝐷𝐶̅̅ ̅̅ Prove: ΔABD ΔABC

For problems 74-77, write a two column proof for the given information.

74. Given: 𝐴𝐵̅̅ ̅̅ is an angle bisector of ∠DAC and 𝐴𝐷̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ Prove: ΔABD ΔABC

75. Given: B is the midpoint of 𝐷𝐶̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ Prove: ΔABD ΔABC

76. Given: B is the midpoint of 𝐷𝐸̅̅ ̅̅ and 𝐴𝐶̅̅ ̅̅ and ∠ABE is a right angle. Prove: ΔABE

ΔCBD

77. Given: 𝐷𝐵̅̅ ̅̅ is the angle bisector of ∠ADC and 𝐴𝐷̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ . Prove: ΔABD ΔCBD

For problems 78-81, determine if the two triangles are congruent, using the distance formula.

Leave all your answers in simplest radical form (simplify all radicals, no decimals).

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80. ΔABC: A(-1, 5), B(-4, 2), C(2, -2) and ΔDEF: D(7, -5), E(4, 2), F(8, -9)

81. ΔABC: A(-8, -3), B(-2, -4), C(-5, -9) and ΔDEF: D(-7, 2), E(-1, 3), F(-4, 8)

82. Construct a triangle with sides of length 5cm, 3cm, and 2cm.

For problems 83-96, determine if the triangles are congruent. If they are, write the congruence

statement and which congruence postulate or theorem you used.

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For problems 97-101, use the picture and the given information below.

Given: 𝐷𝐵̅̅ ̅̅ ⊥ 𝐴𝐶̅̅ ̅̅ , 𝐷𝐵̅̅ ̅̅ is the angle bisector of ∠CDA

97. From 𝐷𝐵̅̅ ̅̅ ⊥ 𝐴𝐶̅̅ ̅̅ , which angles are congruent and why?

98. Because 𝐷𝐵̅̅ ̅̅ is the angle bisector of ∠CDA, what two angles are congruent?

99. From looking at the picture, what additional piece of information are you given? Is this

enough to prove the two triangles are congruent?

100. Write a two column proof to prove ΔCDB ΔADB.

101. What would your reason for ∠C ∠A?

For problems 102-106, use the picture below and the given information.

Given: 𝐿𝑃̅̅̅̅ ∥ 𝑁𝑂̅̅ ̅̅ , 𝐿𝑃̅̅̅̅ ≅ 𝑁𝑂̅̅ ̅̅

102. From 𝐿𝑃̅̅̅̅ ∥ 𝑁𝑂̅̅ ̅̅ , which angles are congruent and why?

103. From looking at the picture, what additional piece of information can you

conclude?

104. Write a two column proof to prove ΔLMP ΔOMN.

105. What would your reason be for 𝐿𝑀̅̅ ̅̅ ≅ 𝑀𝑂̅̅ ̅̅ ̅?

106. Fill in the blanks for the proof below. Use the given and picture from above.

Prove: M is the midpoint of 𝑃𝑁̅̅ ̅̅

For problems 107-112, determine the additional piece of information needed to show the two

triangles are congruent by the given postulate.

107. AAS

108. ASA

109. ASA

110. AAS

111. HL

112. SAS

For problems 113 and 114, write a two column proof.

113. Given: 𝑆𝑉̅̅̅̅ ⊥ 𝑊𝑈̅̅ ̅̅ ̅ and Ti is the midpoint of both 𝑆𝑉̅̅̅̅ and 𝑊𝑈̅̅ ̅̅ ̅. Prove: 𝑊𝑆̅̅̅̅̅ ≅ 𝑈𝑉̅̅ ̅̅

114. Given: ∠K ∠T and 𝐸𝐼̅̅ ̅ is the angle bisector of ∠KET. Prove: 𝐸𝐼̅̅ ̅ is the angle

bisector of ∠KIT.

For problems 115-124, find the measure of x and/or y.

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125. Is ΔABC isosceles? Explain your reasoning.

126. ΔEQG is an equilateral triangle. If 𝐸𝑈̅̅ ̅̅ bisects ∠LEQ, find:

a. m∠EUL

b. m∠UEL

c. m∠ELQ

d. If EQ = 4, find LU.

For problems 127-131, determine if the following statements are ALWAYS, SOMETIMES, or

NEVER true. Explain your reasoning.

127. Base angles of an isosceles triangle are congruent.

128. Base angles of an isosceles triangle are complementary.

129. Base angles of an isosceles triangle can be equal to the vertex angle.

130. Base angles of an isosceles triangle can be right angles.

131. Base angles of an isosceles triangle are acute.

132. In the diagram below, 𝑙1 ∥ 𝑙2. Find all of the lettered angles.

For problems 133, fill in the blanks in the proof.

133. Given: Isosceles ΔCIS, with base angles ∠C and ∠S, and 𝐼𝑂̅̅ ̅ is the angle bisector

of ∠CIS. Prove: 𝐼𝑂̅̅ ̅ is the perpendicular bisector of 𝐶𝑆̅̅̅̅ .

For problems 134-136, write a two column proof.

134. Given: Equilateral ΔRST with 𝑅𝑇̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅ ≅ 𝑆𝑇̅̅̅̅ . Prove: ΔRST is equiangular.

135. Given: Isosceles ΔICS with ∠C and ∠S, and 𝐼𝑂̅̅ ̅ is the perpendicular bisector of

𝐶𝑆̅̅̅̅ . Prove: 𝐼𝑂̅̅ ̅ is the angle bisector of ∠CIS.

136. Given: Isosceles ΔABC with base angles ∠B and ∠C. Isosceles ΔXYZ with base

angles ∠Y and ∠Z. ∠C ∠Z and 𝐵𝐶̅̅ ̅̅ ≅ 𝑌𝑍̅̅̅̅ . Prove: ΔABC ΔXYZ