### Video Transcript

In the figure below, π΄π΅πΆ is an equilateral triangle, π·π΅ equals π·πΆ, and the measure of angle π΅π·πΆ equals 112 degrees. Find the measure of angle π΄π΅π·.

So as we start this question, letβs remind ourselves of the notation that we use for angles. When weβre talking about the angle thatβs labeled π΄π΅π·, this means that weβre looking for the angle thatβs made at point π΅ and thatβs created by the lines π΄π΅ and π΅π·. So here we have the angle π΄π΅π· marked in pink on the diagram. So if we look at the large triangle on the outside, triangle π΄π΅πΆ, weβre told that itβs an equilateral triangle. That means itβs got three equal sides and three equal angles, which are all 60 degrees each.

If we have a look at the triangle π΅πΆπ·, weβre told that the line π·π΅ is equal to π·πΆ. If we have a triangle that has two equal length sides, this means that the triangle is isosceles. And since triangle π΅πΆπ· is isosceles, this means it must also have two equal angles. So this means that in our triangle π΅πΆπ·, we can say that the measure of angle π·πΆπ΅ is equal to the measure of angle π·π΅πΆ. Letβs 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 π΅π·πΆ 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 π·πΆπ΅ plus the measure of angle π·π΅πΆ plus the measure of angle π΅π·πΆ equals 180 degrees. Letβs create the value π₯ which represents the measure of angle π·πΆπ΅. Since we know that thereβs the other equal angle, angle π·π΅πΆ, we can also say that this is equal to π₯ as well. We know that the measure of angle π΅π·πΆ equals 112. So we can substitute this into our equation, giving us π₯ plus π₯ plus 112 equals 180.

And we can collect our π₯s to give us two π₯ plus 112 equals 180. Then we need to rearrange our equation to give us π₯ by itself. And we could start by subtracting 112 from both sides of the equation, giving us two π₯ equals 180 take away 112. And so, two π₯ equals 68 degrees. To find π₯ by itself then, we divide both sides of our equation by two. Which gives us π₯ equals 68 divided by two, which we can write as π₯ equals 34 degrees.

So now we know that triangle π΅πΆπ· 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 π΄π΅π·, which we still havenβt quite found out yet. So letβs return to the fact that triangle π΄π΅πΆ 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 π΄π΅πΆ 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 π΄π΅π·, our small pink angle, must be equal to the measure of angle π΄π΅πΆ subtract the measure of angle π·π΅πΆ. And plugging in the angles that we know would give us the measure of angle π΄π΅π· is equal to 60 degrees subtract 34 degrees. Which we can evaluate as 26 degrees. So our final answer is, the measure of angle π΄π΅π· equals 26 degrees.