Let's look at the relevant parts of the diagram. We are given the measure of one of the angles formed by two intersecting lines, and we are asked to find the measure of an angle not adjacent to this angle. Let's call the given angle ∠ 8. The information on the diagram tells us that m ∠ 8=53^(∘).
Since ∠ 8 and ∠ 7 are nonadjacent angles formed by two intersecting lines, they are vertical angles.
According to the Vertical Angles Theorem, vertical angles are congruent.
∠ 7≅ ∠ 8
By the definition of congruence, this means that these angles have the same measure.
m ∠ 7=m ∠ 8
It is given that m ∠ 8=53^(∘), so according to the Transitive Property of Equality, we can replace m ∠ 8 with 53^(∘).
m ∠ 7=53^(∘)
Thus, we proved that the measure of ∠ 7 is 53^(∘).
