By the Vertical Angles Congruence Theorem, ∠ 2 and ∠ 4 are congruent.
m∠ 2 =m∠ 4
As m∠ 2 =59^(∘), it must follow that m∠ 4 =59^(∘). 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.
