Looking at the figure, we notice three things we need to figure out the remaining angles.
∠ 1 and ∠ 3 are vertical angles.
∠ 1 is a supplementary angle to both ∠ 2 and ∠ 4.
∠ 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.
