Video Transcript
Given that π΄π΅ equals 92
centimeters, π΄πΆ equals 91 centimeters, and πΆπΈ equals π΅π·, choose the correct
relationship between the measure of angle π΄πΈπ· and the measure of angle
π΄π·πΈ. Option (A) the measure of angle
π΄πΈπ· is greater than the measure of angle π΄π·πΈ. Option (B) the measure of angle
π΄πΈπ· is equal to the measure of angle π΄π·πΈ. Or is it option (C) the measure of
angle π΄πΈπ· is less than the measure of angle π΄π·πΈ?
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΄πΆ is 91 centimeters. We can see the fact that πΆπΈ
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΄πΈπ· and π΄π·πΈ.
We can see that the two angles
whose measures we want to compare are interior angles in triangle π΄π·πΈ. 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΄πΈπ· and π΄π·πΈ by comparing the lengths of the sides opposite
these angles in triangle π΄π·πΈ.
We want to compare the lengths of
π΄πΈ and π΄π·. We can compare these lengths by
noting that line segments πΆπΈ and π΅π· are the same length. Letβs say π₯ centimeters. This means that π΄π· has length 92
minus π₯ centimeters and π΄πΈ 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΄πΈ.
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΄πΈπ· is
greater than the measure of angle π΄π·πΈ, which is option (A).