Video Transcript
True or False: In the given figure,
if the measure of angle 𝐵 is equal to the measure of angle 𝐶 and the line segment
between 𝐴 and 𝐷 is a bisector of angle 𝐴, then triangle 𝐴𝐵𝐷 is congruent to
triangle 𝐴𝐷𝐶.
In this question, we are given a
geometric figure and asked to determine if two triangles in the figure are
congruent, which means that they would have corresponding sides of the same length
and corresponding angles of the same measure. We can do this by considering the
congruency criteria for triangles. Let’s begin by adding the given
information onto the sketch.
First, we are told that the measure
of angles 𝐵 and 𝐶 are equal. Second, we are told that line
segment 𝐴𝐷 bisects the angle at 𝐴. This means that it splits the angle
into two angles of equal measure as shown.
There are a few different ways to
consider the congruency of the two triangles. One way is to note that triangle
𝐴𝐵𝐶 is isosceles since it has two angles of equal measure. We can then recall that the
isosceles triangle theorem tells us that the sides opposite the congruent angles in
an isosceles triangle have the same length. So, 𝐴𝐵 is equal to 𝐴𝐶 as
shown.
We have now shown that triangles
𝐴𝐵𝐷 and 𝐴𝐶𝐷 have two angles of equal measure and the included side of these
corresponding angles have the same length. This is enough to prove that the
two triangles are congruent using the angle-side-angle congruence criterion. Hence, the answer is true.
