We want to use the given figure to determine which of the angles has the greatest measure. In this exercise we will consider angles ∠ 3, ∠ 9, and ∠ 14. Let's begin with analyzing the given diagram and marking the angles we are interested in.
As we can see, ∠ 3 is an exterior angle, with ∠ 9 and the sum of ∠ 11 and ∠ 14 as its corresponding remote interior angles. According to the Exterior Angle Inequality Theorem, ∠ 3 is greater than either of its corresponding remote angles. Therefore, m∠ 3 > m∠ 9 and m∠ 3> m∠ 11+ m∠ 14. The second inequality results in the following inequalities and conclusions.
m∠ 3> m∠ 11+ m∠ 14
⇓
m∠ 3> m∠ 11 and m∠ 3>m∠ 14
The obtained inequalities tell us that the measure of angle ∠ 3 is greater than both the measure of angle ∠ 9 and the measure of angle ∠ 14. Therefore, we can finally conclude that angle ∠ 3 has the greatest measure out of these three angles.
