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

Given that 𝐴𝐵 = 92 cm, 𝐴𝐶 = 91 cm, and 𝐶𝐸 = 𝐵𝐷, choose the correct relationship between 𝑚∠𝐴𝐸𝐷 and 𝑚∠𝐴𝐷𝐸. [A] 𝑚∠𝐴𝐸𝐷 > 𝑚∠𝐴𝐷𝐸 [B] 𝑚∠𝐴𝐸𝐷 = 𝑚∠𝐴𝐷𝐸 [C] 𝑚∠𝐴𝐸𝐷 < 𝑚∠𝐴𝐷𝐸

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Video Transcript

Given that 𝐴𝐵 equals 92 centimeters, 𝐴𝐶 equals 91 centimeters, and 𝐶𝐸 equals 𝐵𝐷, choose the correct relationship between the measure of angle 𝐴𝐸𝐷 and the measure of angle 𝐴𝐷𝐸. Option (A) the measure of angle 𝐴𝐸𝐷 is greater than the measure of angle 𝐴𝐷𝐸. Option (B) the measure of angle 𝐴𝐸𝐷 is equal to the measure of angle 𝐴𝐷𝐸. Or is it option (C) the measure of angle 𝐴𝐸𝐷 is less than the measure of angle 𝐴𝐷𝐸?

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 𝐴𝐵 is 92 centimeters and 𝐴𝐶 is 91 centimeters. We can see the fact that 𝐶𝐸 equals 𝐵𝐷 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 𝐴𝐸𝐷 and 𝐴𝐷𝐸.

We can see that the two angles whose measures we want to compare are interior angles in triangle 𝐴𝐷𝐸. 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 𝐴𝐸𝐷 and 𝐴𝐷𝐸 by comparing the lengths of the sides opposite these angles in triangle 𝐴𝐷𝐸.

We want to compare the lengths of 𝐴𝐸 and 𝐴𝐷. We can compare these lengths by noting that line segments 𝐶𝐸 and 𝐵𝐷 are the same length. Let’s say 𝑥 centimeters. This means that 𝐴𝐷 has length 92 minus 𝑥 centimeters and 𝐴𝐸 has length 91 minus 𝑥 centimeters. We know that 𝑥 is nonnegative since it represents a length. This means that 92 minus 𝑥 is greater than 91 minus 𝑥. So 𝐴𝐷 is a longer side than 𝐴𝐸.

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 𝐴𝐸𝐷 is greater than the measure of angle 𝐴𝐷𝐸, which is option (A).