Video Transcript
From the figure, what is the
relationship between the measure of angle 𝐴𝐵𝐷 and the measure of angle
𝐴𝐶𝐷?
In this question, we’re given a
figure containing two triangles that share a side, one of which is isosceles. We are also given two side lengths
of a triangle. We need to use this to compare the
measures of two of the angles in the figure.
To do this, let’s start by marking
the two angles whose measures we want to compare on the diagram. To compare the measures of these
two angles, we will compare the measures of the part of the two angles in each
triangle.
First, we see that triangle 𝐷𝐶𝐵
is isosceles. We know that the angles opposite
the sides of the same length are congruent. So we can add this information onto
our diagram. Since angles 𝐷𝐶𝐵 and 𝐷𝐵𝐶 have
the same measure, we can compare the measures of the two angles that we have
highlighted by comparing the measures of the angles in triangle 𝐴𝐵𝐶.
We can compare the measures of
angles in a triangle from their side lengths by using the angle comparison theorem
in triangles. This tells us that an angle
opposite a longer side in a triangle must have larger measure than an angle opposite
a shorter side in the same triangle. From the diagram, we can see that
the angle at 𝐵 in the triangle is opposite a shorter side than the side opposite
angle 𝐶. So the angle at 𝐵 in the triangle
has larger measure than the angle at 𝐶 in the triangle.
We can then add the measures of the
congruent angles in the isosceles triangle to each side of the inequality to get the
following inequality. On the left-hand side of the
inequality, we have the measure of angle 𝐴𝐵𝐶 plus the measure of angle
𝐷𝐵𝐶. We can see that this is the sum of
the measures of the angles that make angle 𝐴𝐵𝐷. Similarly, on the right-hand side
of the inequality, we are adding the measures of the two angles that combine to make
angle 𝐴𝐶𝐷. This allows us to rewrite the
inequality as the measure of angle 𝐴𝐵𝐷 is greater than the measure of angle
𝐴𝐶𝐷.
