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).
