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.
