Chapter 3

Geometric Proofs with Triangles

In geometry, you're frequently asked to prove something. In this chapter, you're given specific information and asked to prove specific information about triangles. You do this by using various geometric properties, postulates, and definitions to generate new statements that will lead you toward the information you're looking to prove true.

The Problems You'll Work On

In this chapter, you see a variety of problems involving geometric proofs:

• Using SAS, SSS, ASA, and AAS to prove triangles congruent
• Showing that corresponding parts of congruent triangles are congruent
• Formulating a geometric proof with overlapping triangles
• Using your knowledge of quadrilaterals to complete a geometric proof
• Completing indirect proofs

What to Watch Out For

Remember the following tips as you work through this chapter:

• The statement that needs to be proven has to be the last statement of the proof. It can't be used as a given statement.
• You must use all given information to formulate the proof. Each given should be used separately to draw its own conclusion.
• If you've used all your given information and still require more to prove the triangles congruent, look for the reflexive property or a pair of vertical angles.
• After you find angles or segments congruent, mark them in your diagram. The markings make it easier for you to see what other information you need to complete the proof.
• To prove parts of a triangle congruent, you'll first need to prove that the triangles are congruent to each other using the proper triangle congruence theorems.

Triangle Congruence Theorems

86-102 Use your knowledge of SAS, ASA, SSS, and AAS to solve the problem.

86. What method can you use to prove these two triangles congruent?

87. What method can you use to prove these two triangles congruent?

88. What method can you use to prove these two triangles congruent?

89. What method can you use to prove

90. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the SSS method?

91. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the SAS method?

92. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the AAS method?

93. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the ASA method?

94. Which pair of segments or angles would need to be proved congruent in order to prove these triangles congruent using the SSS method?

95. Which pair of segments or angles would need to be proved congruent in order to prove the triangles congruent using the SAS method?

96. Given: bisects and E is the midpoint of . Is it possible to prove using only the given information and the reflexive property?

97. Given: bisects and bisects . Which method of triangle congruence would you use to prove

98. Given: and . Which method of triangle congruence would you use to prove

99. Given: is the altitude drawn to , and bisects . Which method of triangle congruence would you use to prove

100. Given: and . Which method of triangle congruence would you use to prove

101. Given: , and . Which method of triangle congruence would you use to prove

102. Given: Quadrilateral , and . Which method of triangle congruence would you use to prove

Completing Geometric Proofs Using Triangle Congruence Theorems

103-107 Use the following figure to answer each question.

Given: and bisect each other at B.

Prove:

Statements

Reasons

1. and bisect each other at B.

1. Given

2. and

2.

3.

3. Intersecting lines form vertical angles.

4.

4.

5.

5.

6.

6.

103. What is the reason for Statement 2?

104. What is the statement for Reason 3?

105. What is the reason for Statement 4?

106. What is the reason for Statement 5?

107. What is the reason for Statement 6?

108-111 Use the following figure to answer each question.

Given: ,

and

Prove:

Statements

Reasons

1.

1. Given

2.

2.

3.

or

3.

4.

4.

5.

5.

108. What is the reason for Statement 2?

109. What is the reason for Statement 3?

110. What is the reason for Statement 4?

111. What is the reason for Statement 5?

112-116 Use the following figure to answer each question.

Given: and

Prove:

Statements

Reasons

1. and

1. Given

2.

2. When two parallel lines are cut by a transversal, alternate interior angles are formed.

3.

3.

4.

4. Reflexive property

5.

5.

6.

6.

112. What is the statement for Reason 2?

113. What is the reason for Statement 3?

114. What is the statement for Reason 4?

115. What is the reason for Statement 5?

116. What is the reason for Statement 6?

117 Complete the following proof.

Given: , and M is the midpoint of .

Prove:

Overlapping Triangle Proofs

118-120 Use the following figure to answer each question regarding overlapping triangles.

Given: and

Prove:

Statements

Reasons

1. and

1. Given

2.

2.

3.

3.

4.

4.

118. What is the reason for Statement 2?

119. What is the reason for Statement 3?

120. What is the reason for Statement 4?

121-125 Use the following figure to answer each question regarding overlapping triangles.

Given: , and

Prove:

Statements

Reasons

1. , , and

1. Given.

2.

2. Perpendicular lines form right angles.

3.

3.

4.

4.

5.

5.

6.

6.

121. What is the statement for Reason 2?

122. What is the reason for Statement 3?

123. What is the reason for Statement 4?

124. What is the reason for Statement 5?

125. What is the reason for Statement 6?

126-130 Use the following figure to answer each question regarding overlapping triangles.

Given: and

Prove:...