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

Use <, =, or > to fill in the blank: πβ π΄πΆπ· οΌΏ πβ π΄π΅πΆ.

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### Video Transcript

Use the less than, equals, or greater than symbol to fill in the blank. The measure of angle π΄πΆπ· what the measure of angle π΄π΅πΆ.

Letβs begin by marking the two angles we want to compare on the diagram. Thatβs angles π΄πΆπ· and π΄π΅πΆ. At a glance, the angle at πΆ appears larger than the angle at π΅. But to answer the question, we really need to prove that this is the case. We can do this by noting that weβre given a pair of parallel lines and that π΅πΆ 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 π΄π΅πΆ is equal to the measure of angle π΅πΆπ·. Now, the angle at πΆ, thatβs angle π΄πΆπ·, is equal to the sum of the measures of angles π΄πΆπ΅ and π΅πΆπ·. If we substitute the measure of angle π΄π΅πΆ for the measure of angle π΅πΆπ·, we have the measure of angle π΄πΆπ· equal to the measure of angle π΄πΆπ΅ plus the measure of angle π΄π΅πΆ.

Recalling that itβs the measures of angles π΄πΆπ· and π΄π΅πΆ that weβre concerned with, and since we know that the measure of angle π΄πΆπ΅ is greater than zero, it must be the case that the measure of angle π΄πΆπ· is greater than that of angle π΄π΅πΆ. Hence, the symbol we require to fill in the blank is the greater than symbol.