# Question Video: Identifying the Relation between Two Given Angles Mathematics • 11th Grade

From the figure, what is the relationship between 𝑚∠𝐴𝐵𝐷 and 𝑚∠𝐴𝐶𝐷?

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### 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.