Exercise 9 Page 111

Looking at the figure, we notice three things we need to figure out the remaining angles.

1. ∠ 4 and ∠ 2 are vertical angles.
2. ∠ 2 is supplementary to both ∠ 1 and ∠ 3.
3. ∠ 1 and ∠ 3 are vertical angles.

By the Vertical Angles Congruence Theorem, ∠ 4 and ∠ 2 are congruent. m∠ 4 =m∠ 2 As m∠ 2 =34^(∘), it must follow that m∠ 4 =34^(∘). Supplementary angles have measures that sum to 180^(∘). Additionally, since we also know that ∠ 1 and ∠ 3 are vertical angles, we can write the following three equations. m∠ 2+m∠ 3&=180 m∠ 2+m∠ 1&=180 m∠ 1&=m∠ 3 By solving the first equation for m∠ 3, we also figure out the measure of ∠ 1.

m∠ 2+m∠ 3=180

Substitute

m∠ 2= 34

34+m∠ 3=180

SubEqn

LHS-34=RHS-34

m∠ 3=146

Let's summarize what we have found. m∠ 1&=146^(∘) m∠ 3&=146^(∘) m∠ 4&=34^(∘)