We are given the following diagram of different paths in a park. We are told that the bike path and the horse riding path are parallel. The hiking trail intersects the two paths in one part of the park. With this in mind, we want to find the measures of ∠ 1 and ∠ 2. Let's look at the diagram.
Angles that lie on the inside of the parallel lines and on opposite sides of a transversal.
In this case, the bike path and the horse riding path are parallel lines which are cut by a transversal — the hiking trail. The angle that measures 72^(∘) lies on the inside of the parallel paths. Let's find its alternate interior angle.
The angle that we named ∠ 3 and the 72^(∘) angle are alternate interior angles, so they are congruent. Congruent angles have the same measure, so the measure of ∠ 3 is 72^(∘). Next, notice that ∠ 2 and ∠ 3 form a straight line.
When two angles form a straight lines, they are called a linear pair. Angles in a linear pair are supplementary.
Supplementary Angles
Angles whose measures add up to 180^(∘).
With this information, we can find the measure of ∠ 2.
