Exercise 30 Page 502

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

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

By the Vertical Angles Congruence Theorem, ∠ 1 and ∠ 3 are congruent. Therefore we have that m∠ 1 =m∠ 3. As m∠ 1 =76^(∘) it must follow that m∠ 3 =76^(∘). Supplementary angles have a measure that adds up to 180^(∘). Additionally, since we also know that ∠ 2 and ∠ 4 are vertical angles, we can write the following three equations: m∠ 1+m∠ 2&=180 m∠ 1+m∠ 4&=180 m∠ 2&=m∠ 4 By solving the first equation for m∠ 2 we also figure out the measure of ∠ 4.

Let's summarize what we have found: m∠ 2&=104^(∘) m∠ 3&=76^(∘) m∠ 4&=104^(∘)