Exercise 5 Page 304

Let's begin by looking at the given information and the desired outcome of the proof. Given:& ∠ 2 ≅ ∠ 6 Prove:& ∠ 4 ≅ ∠ 8 From the diagram below, we can see that ∠ 2 and ∠ 4 as well as ∠ 6 and ∠ 8 are supplementary.

Because supplementary angles always add to be 180, this gives us two important equations. m∠ 2 + m∠ 4 = 180 & (I) m∠ 6 + m∠ 8 = 180 & (II) Moreover, since ∠ 2 ≅ ∠ 6, we have that m∠ 2= m∠ 6, which allows us to rewrite the system above. m∠ 6 + m∠ 4 = 180 & (I) m∠ 6 + m∠ 8 = 180 & (II) Now, we can use the Substitution Method to solve the system.

m∠ 6 + m∠ 4 = 180 & (I) m∠ 6 + m∠ 8 = 180 & (II)

Substitute

(II): 180= m∠ 6 + m∠ 4

m∠ 6 + m∠ 4 = 180 m∠ 6 + m∠ 8 = m∠ 6 + m∠ 4

SubEqn

(II): LHS-m∠ 6=RHS-m∠ 6

m∠ 6 + m∠ 4 = 180 m∠ 8 = m∠ 4

RearrangeEqn

(II): Rearrange equation

m∠ 6 + m∠ 4 = 180 m∠ 4 = m∠ 8

The second equation implies that ∠ 4 ≅ ∠ 8. We can summarize the steps we've taken in a two-column proof table.

Statements	Reasons
1. ∠ 2 ≅ ∠ 6	1. Given
2. m∠ 2 = m∠ 6	2. Definition of ≅
3. m∠ 2 + m∠ 4=180 & (I) m∠ 6 + m∠ 8=180 & (II)	3. Supplement Theorem
4. m∠ 2 + m∠ 4 = m∠ 6 + m∠ 8	4. Substitution Method
5. m∠ 6 + m∠ 4 = m∠ 6 + m∠ 8	5. Substitution
6. m∠ 4 = m∠ 8	6. Subtraction Property of Equality
7. ∠ 4 ≅ ∠ 8	7. Definition of ≅