Exercise 24 Page 431

We want to use the given figure to determine which of the angles has the greatest measure. In this exercise we will consider angles ∠ 1, ∠ 5, and ∠ 6. Let's begin with analyzing the given diagram and marking the angles we are interested in.

Notice that ∠ 5 and ∠ 6 are interior angles, while ∠ 1 is an exterior angle of the same triangle. Moreover, because ∠ 5 and ∠ 6 do not share a vertex or corner of a triangle with ∠ 1, they are corresponding remote angles of ∠ 1. Now, let's review the Exterior Angle Inequality Theorem.

According to this theorem, the measure of angle ∠ 1 is greater than both the measure of angle ∠ 5 and the measure of angle ∠ 6. Therefore, ∠ 1 has the greatest measure out of these three angles.