Video: Finding the Unknown Measure of an Angle in an Isosceles Triangle Using Its Properties

Video Transcript

In the figure below, given that 𝐴𝐵 is equal to 𝐵𝐶, 𝐴𝐷 is equal to 𝐶𝐷, and the measure of angle 𝐵𝐴𝐷 is equal to 108 degrees, find the measure of angle 𝐴𝐷𝐶.

Well the first thing we want to do is mark on the information we have. And the first bit of information we have is the fact that measure of the angle 𝐵𝐴𝐷 is equal to 108 degrees. So I’ve marked this on on our diagram. And next, I’ve marked on the angle at 𝐴𝐷𝐶 because this is the angle that we’re trying to find.

So now, if we take a look at some information we’ve got from the question and the markings on the diagram, we can see that we’re dealing with two isosceles triangles. And that’s because we’re told that 𝐴𝐵 is equal to 𝐵𝐶 and 𝐴𝐷 is equal to 𝐶𝐷, and we’ve also got these lines on the diagram that denote this.

So the first thing that we can draw from the fact that we’re dealing with isosceles triangles is that measure of the angle 𝐵𝐴𝐶 is equal to the measure of the angle 𝐴𝐶𝐵. And this is because they’re both the base angles of an isosceles triangle. So therefore, these are going to be equal.

So therefore, what I’m gonna do is set up an equation. And to do that, I’m gonna call the measure of angle 𝐵𝐴𝐶 and the measure of angle 𝐴𝐶𝐵 𝑥. Because as we’ve already stated, they are the same. Our equation is two 𝑥 plus 36 is equal to 180. And that’s because the angles in a triangle sum to 180 degrees. And the three angles in this triangle are 𝑥, 𝑥, and 36. As you can see, I’ve given reasoning for every step of my working, and you must do that when you’re dealing with angle problems.

So now what I’m going to do is solve the equation to find 𝑥. And the first step is to subtract 36 from each side of the equation. And when I do that, I get two 𝑥 is equal to 144. And then the next step is to divide each side of the equation by two. And when I do that, I get a value of 𝑥 of 72 degrees.

Okay, great. So now we know angles 𝑥, or the base angles of the triangle. So now what we’re going to do is look at the triangle 𝐴𝐶𝐷. And in the triangle 𝐴𝐶𝐷, we can see, again, it’s isosceles. So therefore, the measure of angle 𝐶𝐴𝐷 is equal to the measure of angle 𝐴𝐶𝐷, because again, they’re the base angles of an isosceles triangle.

This time, I’m going to denote the angles as 𝑦. And we can work out the angle at 𝑦 by subtracting 72 from 108. And that’s because in the question, we’re told that the measure of angle 𝐵𝐴𝐷 is 108. And we’ve calculated the measure of angle 𝐵𝐴𝐶 as 72 degrees. So therefore, we can take these away to find the measure of angle 𝐶𝐴𝐷, or 𝑦. So therefore, when we do this, we can say that the measure of angle 𝐶𝐴𝐷 and the measure of angle 𝐴𝐶𝐷, or just 𝑦, is equal to 36 degrees.

But, have we solved the problem? Well, no, we’ve just found 𝑦. What we’re trying to find is the measure of angle 𝐴𝐷𝐶. And we need to use our isosceles triangle property once again to help us to find this. So once again, we can set up an equation. We’ve got the measure of angle 𝐴𝐷𝐶 plus two 𝑦 is equal to 180. And again, this is because the angles in a triangle sum to 180 degrees.

So then, if we substitute in our value for 𝑦, we get the measure of angle 𝐴𝐷𝐶 plus two multiplied by 36 is equal to 180. So therefore, we get the measure of angle 𝐴𝐷𝐶 plus 72 is equal to 180. So just one more step, we need to subtract 72 from each side of the equation. And when we do this, we get to our final answer. And that is that the measure of angle 𝐴𝐷𝐶 is equal to 108 degrees.