Question Video: Using the Triangle Inequality Theorem to Compare Angle Measures Mathematics

Video Transcript

Use the less than, equals, or greater than symbol to fill in the blank. The measure of angle 𝐴𝐶𝐷 what the measure of angle 𝐴𝐵𝐶.

Let’s begin by marking the two angles we want to compare on the diagram. That’s angles 𝐴𝐶𝐷 and 𝐴𝐵𝐶. At a glance, the angle at 𝐶 appears larger than the angle at 𝐵. But to answer the question, we really need to prove that this is the case. We can do this by noting that we’re given a pair of parallel lines and that 𝐵𝐶 is a transversal of these parallel lines.

Now, recalling that alternate interior angles in a transversal of parallel lines are congruent, we have that the measure of angle 𝐴𝐵𝐶 is equal to the measure of angle 𝐵𝐶𝐷. Now, the angle at 𝐶, that’s angle 𝐴𝐶𝐷, is equal to the sum of the measures of angles 𝐴𝐶𝐵 and 𝐵𝐶𝐷. If we substitute the measure of angle 𝐴𝐵𝐶 for the measure of angle 𝐵𝐶𝐷, we have the measure of angle 𝐴𝐶𝐷 equal to the measure of angle 𝐴𝐶𝐵 plus the measure of angle 𝐴𝐵𝐶.

Recalling that it’s the measures of angles 𝐴𝐶𝐷 and 𝐴𝐵𝐶 that we’re concerned with, and since we know that the measure of angle 𝐴𝐶𝐵 is greater than zero, it must be the case that the measure of angle 𝐴𝐶𝐷 is greater than that of angle 𝐴𝐵𝐶. Hence, the symbol we require to fill in the blank is the greater than symbol.