Video Transcript
In the figure, 𝐴𝐵 is greater than
𝐵𝐶 and 𝐴𝐷 is greater than 𝐷𝐶. Which of the following is true? Option (A) the measure of angle
𝐵𝐶𝐷 is less than the measure of 𝐵𝐴𝐷. Option (B) the measure of angle
𝐵𝐶𝐷 is equal to the measure of angle 𝐵𝐴𝐷. Option (C) the measure of angle
𝐵𝐶𝐷 is greater than the measure of angle 𝐵𝐴𝐷. Option (D) the measure of angle
𝐵𝐶𝐴 is less than the measure of angle 𝐵𝐴𝐶. Or is it option (E) the measure of
angle 𝐴𝐶𝐷 is greater than the measure of angle 𝐴𝐵𝐶?
In this question, we are given a
figure, and we are given the length comparison of two sides in the figure. We want to use this information to
determine the correct comparison of the measures of two angles in the figure. Since we are given a comparison of
the lengths of sides in the figure and we want to compare the measures of angles, we
can start by recalling the angle comparison theorem in triangles. This tells us that if one side in a
triangle is longer than another side in a triangle, then it must be opposite an
angle of larger measure than the other angle in the triangle.
Let’s start by applying this
property to the first two sides. We can start by highlighting the
longer of these sides, 𝐴𝐵, in orange and the shorter side, 𝐵𝐶, in pink as
shown. We can see that these are two sides
in triangle 𝐴𝐵𝐶. We can now apply the angle
comparison theorem to this triangle. We know that the angle opposite the
longer side will have larger measure. So, we can highlight the angle with
larger measure in orange and the one with smaller measure in pink. We have that the measure of angle
𝐵𝐶𝐴 is greater than the measure of angle 𝐵𝐴𝐶.
We can apply this same process with
the other pair of sides. We will once again highlight the
longer side, 𝐴𝐷, in orange, and the shorter side, 𝐷𝐶, in pink as shown. We can then apply the angle
comparison theorem to triangle 𝐴𝐶𝐷. We see that the angle opposite the
longer side must have the larger measure. So we have that the measure of
angle 𝐴𝐶𝐷 is greater than the measure of angle 𝐶𝐴𝐷.
We can now see that the two angles
that make up the internal angle at 𝐶 have larger measure than the two angles that
make up the internal angle at 𝐴. So the measure of the angle at 𝐶
is greater than the measure of the angle at 𝐴. We can write these angles out in
full using the vertices of the quadrilateral to obtain that the measure of angle
𝐵𝐶𝐷 is greater than the measure of angle 𝐵𝐴𝐷, which we can see is option
(C).
