# Question Video: Forming and Solving a System of Linear Equations with Two Unknowns Mathematics • 8th Grade

The sizes of the base angles of an isosceles triangle are (9π β 2π)Β° and (5π + 2π)Β°, and the size of the vertex angle is (4π)Β°. Find the values of π and π.

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### Video Transcript

The sizes of the base angles of an isosceles triangle are nine π minus two π degrees and five π plus two π degrees, and the size of the vertex angle is four π degrees. Find the values of π and π.

The question is telling us about the angles in a triangle. Or more specifically, the angles in an isosceles triangle. The sizes of the angles in the triangle are not given to us directly, but theyβre given to us in terms of these unknown variables, π and π. Our job is to work out the values of these two letters.

Letβs think about what we know about isosceles triangles. Isosceles triangles have two sides thatβre equal in length, and they also have two angles thatβre equal in size. These equal angles are referred to as the base angles. Weβre told in our question that the base angles are equal to nine π minus two π degrees and five π plus two π degrees. They must be equal to each other, which means we can form an equation involving π and π.

We have that nine π minus two π is equal to five π plus two π. Now we canβt solve this equation, but we can simplify it. So Iβd like to collect all the πs on one side and all the πs on the other. Adding two π to both sides of the equation gives nine π is equal to five π plus four π. Then subtracting five π from both sides gives four π is equal to four π. Now both sides of this equation have a factor of four, so I can divide by that. And this gives me that π is equal to π. So I still donβt know the values of π and π, but whatever they are, they are equal to each other.

Now letβs look at the other piece of information in the question, which was that the vertex angle is equal to four π. This means I know the size of all three angles in terms of π and π, so I can use the key fact that angles in a triangle sum to 180 degrees to form a second equation involving π and π. If I add together the three angles in this triangle, I have nine π minus two π plus five π plus two π plus four π is equal to 180. Now letβs try to simplify this equation. I have nine π plus five π plus four π, so that gives 18π. Looking at the πs, I can see that I have minus two π plus two π, so they cancel each other out directly. This just gives me then that 18π is equal to 180. Now in order to find the value of π, I can divide by 18. This tells me that π is equal to 10.

So weβve found the value of π, and if we recall that π and π are equal to each other, weβve also found the value of π. Theyβre both equal to 10. And so we have an answer to the problem: both π and π are equal to 10.