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.
