# Question Video: The Triangle Inequality Theorem Mathematics • 11th Grade

Given that π΄π΅ = 92 cm, π΄πΆ = 91 cm, and πΆπΈ = π΅π·, choose the correct relationship between πβ π΄πΈπ· and πβ π΄π·πΈ. [A] πβ π΄πΈπ· > πβ π΄π·πΈ [B] πβ π΄πΈπ· = πβ π΄π·πΈ [C] πβ π΄πΈπ· < πβ π΄π·πΈ

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### Video Transcript

Given that π΄π΅ equals 92 centimeters, π΄πΆ equals 91 centimeters, and πΆπΈ equals π΅π·, choose the correct relationship between the measure of angle π΄πΈπ· and the measure of angle π΄π·πΈ. Option (A) the measure of angle π΄πΈπ· is greater than the measure of angle π΄π·πΈ. Option (B) the measure of angle π΄πΈπ· is equal to the measure of angle π΄π·πΈ. Or is it option (C) the measure of angle π΄πΈπ· is less than the measure of angle π΄π·πΈ?

In this question, we are given some information about the lengths of various sides in a figure and asked to use this information to compare the measures of two angles in the figure.

To do this, we can start by adding the two given lengths onto the diagram. We have that π΄π΅ is 92 centimeters and π΄πΆ is 91 centimeters. We can see the fact that πΆπΈ equals π΅π· is already included on the figure. We can also add the two angles whose measures we want to compare onto the diagram. We want to compare the measures of angles π΄πΈπ· and π΄π·πΈ.

We can see that the two angles whose measures we want to compare are interior angles in triangle π΄π·πΈ. We can recall that the angle comparison theorem for triangles tells us that if one side is longer than another side in a triangle, then the angle opposite the longer side has larger measure. This means that we can compare the measures of angles π΄πΈπ· and π΄π·πΈ by comparing the lengths of the sides opposite these angles in triangle π΄π·πΈ.

We want to compare the lengths of π΄πΈ and π΄π·. We can compare these lengths by noting that line segments πΆπΈ and π΅π· are the same length. Letβs say π₯ centimeters. This means that π΄π· has length 92 minus π₯ centimeters and π΄πΈ has length 91 minus π₯ centimeters. We know that π₯ is nonnegative since it represents a length. This means that 92 minus π₯ is greater than 91 minus π₯. So π΄π· is a longer side than π΄πΈ.

Finally, by the angle comparison theorem in triangles, we know that the interior angle opposite the longer side will have larger measure. So the measure of angle π΄πΈπ· is greater than the measure of angle π΄π·πΈ, which is option (A).