Exercise 31 Page 502

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

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

By the Vertical Angles Congruence Theorem, ∠ 2 and ∠ 4 are congruent. Therefore we have that m∠ 2 =m∠ 4. As m∠ 2 =159^(∘), it must follow that m∠ 4 =159^(∘). Supplementary angles have measures that add up 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.

Let's summarize what we have found: m∠ 1&=21^(∘) m∠ 3&=21^(∘) m∠ 4&=159^(∘).