Author: Tim Heppner
Last saved by: MHE
Company: McGraw-Hill

Geometry	Chapter 4

Lesson 4-6

Example 1  Proof of Theorem
Write a two-column proof of the 	
Isosceles Triangle Theorem.
Given:	△ABC,  ≅ 
Prove:	∠B ≅ ∠C

Proof:
Statements | Reasons
1.	Let T be the midpoint of . | 1.	Every segment has exactly one midpoint.
2.	Draw an auxiliary segment . | 2.	Two points determine a line.
3.	 ≅ | 3.	Midpoint Theorem
4.	 ≅ | 4.	Reflexive Property
5.	 ≅ | 5.	Given
6.	△ATB ≅ △ATC | 6.	SSS
7.	∠B ≅ ∠C | 7.	CPCTC

Example 2  Find the Measure of a Missing Angle
Standardized Test Practice
If  ≅ ,  ≅ , and m∠PMN = 112, 	
what is the measure of ∠NPQ?

A	25	B	34	C	143	D	17

Read the Test Item
△PQN is isosceles with base . Likewise, △MNQ is isosceles with base .

Solve the Test Item
Step 1	The base angles of △MNQ are congruent. Let x = m∠MQN = m∠MNQ.
	m∠MQN + m∠MNQ + PMN	= 180	Angle Sum Theorem
	x + x + 112	= 180	Substitution
	2x + 112	= 180	Add.
	2x	= 68	Subtract 112 from each side.
	x	= 34	Divide each side by 2.
	So, m∠MQN = m∠MNQ = 34

Step 2	∠PQN and ∠MQN form a linear pair. Solve for m∠PQN.
	m∠PQN + m∠MQN	= 180	Linear pairs are supplementary.
	m∠PQN + 34	= 180	Substitution
	m∠PQN	= 146	Subtract 34 from each side.

Step 3	The base angles of △PQN are congruent. Let y represent m∠NPQ and m∠PNQ.
	m∠NPQ + m∠PNQ + m∠PQN	= 180	Angle Sum Theorem
	y + y + 146	= 180	Substitution
	2y + 146	= 180	Add.
	2y	= 34	Subtract 146 from each side.
	y	= 17	Divide each side by 2.

	The measure of ∠NPQ is 17. Choice D is correct.

Example 3  Congruent Segments and Angles
a.	Name two congruent angles.	
	∠NLM is opposite  and ∠NML is opposite,
so ∠NLM ≅ ∠NML.

b.	Name two congruent segments.
	By the converse of the Isosceles Triangle Theorem, 
the sides opposite congruent angles are congruent. 
So,  ≅ .

Example 4  Use Properties of Equilateral Triangles
△QRS is equilateral, and  bisects ∠SQR.	
a.	Find m∠1 and m∠2. 
	Each angle of an equilateral triangle measures 60°.
So, m∠1 + m∠2 = 60. Since the angle was bisected 
m∠1 = m∠2. Thus, m∠1 = m∠2 = 30.

b.	ALGEBRA  Find x.
	m∠QSP + m∠1 + m∠QPS	= 180	Angle Sum Theorem
	60 + 30 + 3x	= 180	Substitution
	90 + 3x	= 180	Add.
	3x	= 90	Subtract 90 from each side.
	x	= 30	Divide each side by 3.