### Video Transcript

In the triangle ๐ด๐ต๐ถ, one of the angles is the arithmetic mean of the other two. Find each angle of the triangle given the difference between the smaller and larger angles is 61 degrees.

Itโs sensible to begin by defining each of the angles in our triangle. Weโll say that each of the three angles in our triangle are equal to ๐ฅ degrees, ๐ฆ degrees, and ๐ง degrees. Before even using any of the information in the question, we know that the sum of the angles is 180 degrees. So we can say that ๐ฅ plus ๐ฆ plus ๐ง must be equal to 180. If we then say that ๐ฅ is the smallest and ๐ง is the largest, it follows that ๐ฆ must be the arithmetic mean of ๐ฅ and ๐ง. In other words, when we find the sum of ๐ฅ and ๐ง and then divide that by two, we end up with the third angle; thatโs ๐ฆ. So ๐ฅ plus ๐ง over two equals ๐ฆ. Letโs neaten this up a little bit and multiply through by two.

When we do, we find that ๐ฅ plus ๐ง equals two ๐ฆ. The other piece of information we have is that the difference between the smaller and larger angles is 61 degrees. So ๐ง minus ๐ฅ must be equal to 61. And weโve created a system of linear equations in three variables, ๐ฅ, ๐ฆ, and ๐ง. So how do we solve each of these equations? Well, ideally, we want to create an equation in just one variable. And so, weโre gonna begin by manipulating the first two equations Iโve labelled one and two. If we subtract two ๐ฆ from both sides of our second equation, we get ๐ฅ minus two ๐ฆ plus ๐ง equal zero. Letโs call this equation three. Notice that in equation one and equation three, we have exactly the same number of ๐ฅs and ๐งs. The only thing that changes is the number of ๐ฆs we have.

So weโre going to subtract equation three from an equation one. This will have the effect of eliminating both ๐ฅ and ๐ง. And weโll just have an equation in terms of ๐ฆ. When we do, weโre left with ๐ฆ minus negative two ๐ฆ equals 180 minus zero. ๐ฆ minus negative two ๐ฆ is three ๐ฆ. And we can solve this equation for ๐ฆ by dividing through by three. And when we do, we obtain ๐ฆ to be equal to 60. So weโve calculated the sides of the middle angle. We need to work out the size of angle ๐ฅ and ๐ง. Weโre going to take the value of ๐ฆ and substitute it into equation two. And when we do, weโll be left with two equations purely in terms of ๐ฅ and ๐ง.

When we do, we get ๐ฅ plus ๐ง equals two times 60. So thatโs 120. So weโve got equation four and five. These are purely in terms of ๐ฅ and ๐ง. Notice, though, that the signs of the ๐ฅs are different in each equation. So weโre going to add these two equations, and that will eliminate the ๐ฅ. We could, alternatively, subtract the equations to eliminate the ๐ง. When we add the expressions on the left-hand side, we get ๐ง minus ๐ฅ plus ๐ฅ plus ๐ง, which is simply two ๐ง. And on the right-hand side, we get 120 plus 61 which is 181.

We solved this equation by dividing through by two. And we see that ๐ง is equal to 90.5. All thatโs left is to calculate the size of angle ๐ฅ. Itโs sensible to go back to either equation four or five. Now, Iโm going to choose equation five because there are no negatives. We get ๐ฅ plus 90.5 equals 120. And then, we solve for ๐ฅ by subtracting 90.5 from both sides, to find that ๐ฅ is equal to 29.5.

In ascending order, our angles are 29.5 degrees, 60 degrees, and 90.5 degrees. Now, we could absolutely check our working out by going back to equation one. We replace ๐ฅ with 29.5, ๐ฆ with 60, and ๐ง with 90.5. Their sum is 180 as required. So we know weโve done our working out correctly. The angles are 29.5, 60, and 90.5 degrees.