In the figure below, 𝐴𝐵𝐶 is an equilateral triangle, 𝐷𝐵 equals 𝐷𝐶, and the measure of angle 𝐵𝐷𝐶 equals 112 degrees. Find the measure of angle 𝐴𝐵𝐷.
So as we start this question, let’s remind ourselves of the notation that we use for angles. When we’re talking about the angle that’s labeled 𝐴𝐵𝐷, this means that we’re looking for the angle that’s made at point 𝐵 and that’s created by the lines 𝐴𝐵 and 𝐵𝐷. So here we have the angle 𝐴𝐵𝐷 marked in pink on the diagram. So if we look at the large triangle on the outside, triangle 𝐴𝐵𝐶, we’re told that it’s an equilateral triangle. That means it’s got three equal sides and three equal angles, which are all 60 degrees each.
If we have a look at the triangle 𝐵𝐶𝐷, we’re told that the line 𝐷𝐵 is equal to 𝐷𝐶. If we have a triangle that has two equal length sides, this means that the triangle is isosceles. And since triangle 𝐵𝐶𝐷 is isosceles, this means it must also have two equal angles. So this means that in our triangle 𝐵𝐶𝐷, we can say that the measure of angle 𝐷𝐶𝐵 is equal to the measure of angle 𝐷𝐵𝐶. Let’s see if we can work out an actual value for each of these angles.
We can use the fact that the angles in a triangle add up to 180 degrees. And since we also know that the angle 𝐵𝐷𝐶 is equal to 112 degrees, we can write an equation which we can then try and solve. Since we know that the three angles in our triangle will add to 180 degrees, this is the same as saying the measure of angle 𝐷𝐶𝐵 plus the measure of angle 𝐷𝐵𝐶 plus the measure of angle 𝐵𝐷𝐶 equals 180 degrees. Let’s create the value 𝑥 which represents the measure of angle 𝐷𝐶𝐵. Since we know that there’s the other equal angle, angle 𝐷𝐵𝐶, we can also say that this is equal to 𝑥 as well. We know that the measure of angle 𝐵𝐷𝐶 equals 112. So we can substitute this into our equation, giving us 𝑥 plus 𝑥 plus 112 equals 180.
And we can collect our 𝑥s to give us two 𝑥 plus 112 equals 180. Then we need to rearrange our equation to give us 𝑥 by itself. And we could start by subtracting 112 from both sides of the equation, giving us two 𝑥 equals 180 take away 112. And so, two 𝑥 equals 68 degrees. To find 𝑥 by itself then, we divide both sides of our equation by two. Which gives us 𝑥 equals 68 divided by two, which we can write as 𝑥 equals 34 degrees.
So now we know that triangle 𝐵𝐶𝐷 has two angles of 34 degrees and an angle of 112 degrees. To check our workings at this point, we could simply add those three angles together and see if we get 180 degrees. So in this question, we were asked to find the measure of angle 𝐴𝐵𝐷, which we still haven’t quite found out yet. So let’s return to the fact that triangle 𝐴𝐵𝐶 is an equilateral triangle. This means that all of the angles will be 60 degrees. In particular, we can say that the measure of angle 𝐴𝐵𝐶 equals 60 degrees.
So to find the missing pink part of our angle, we have 60 degrees, and we can take away 34 degrees. To write this formally, we could say that the measure of angle 𝐴𝐵𝐷, our small pink angle, must be equal to the measure of angle 𝐴𝐵𝐶 subtract the measure of angle 𝐷𝐵𝐶. And plugging in the angles that we know would give us the measure of angle 𝐴𝐵𝐷 is equal to 60 degrees subtract 34 degrees. Which we can evaluate as 26 degrees. So our final answer is, the measure of angle 𝐴𝐵𝐷 equals 26 degrees.