The sizes of the base angles of an isosceles triangle are nine 𝑚 minus two 𝑛 degrees and five 𝑚 plus two 𝑛 degrees, and the size of the vertex angle is four 𝑚 degrees. Find the values of 𝑚 and 𝑛.
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 they’re given to us in terms of these unknown variables, 𝑚 and 𝑛. Our job is to work out the values of these two letters.
Let’s think about what we know about isosceles triangles. Isosceles triangles have two sides that’re equal in length, and they also have two angles that’re equal in size. These equal angles are referred to as the base angles. We’re told in our question that the base angles are equal to nine 𝑚 minus two 𝑛 degrees and five 𝑚 plus two 𝑛 degrees. They must be equal to each other, which means we can form an equation involving 𝑚 and 𝑛.
We have that nine 𝑚 minus two 𝑛 is equal to five 𝑚 plus two 𝑛. Now we can’t solve this equation, but we can simplify it. So I’d like to collect all the 𝑚s on one side and all the 𝑛s on the other. Adding two 𝑛 to both sides of the equation gives nine 𝑚 is equal to five 𝑚 plus four 𝑛. Then subtracting five 𝑚 from both sides gives four 𝑚 is equal to four 𝑛. Now both sides of this equation have a factor of four, so I can divide by that. And this gives me that 𝑚 is equal to 𝑛. So I still don’t know the values of 𝑚 and 𝑛, but whatever they are, they are equal to each other.
Now let’s look at the other piece of information in the question, which was that the vertex angle is equal to four 𝑚. This means I know the size of all three angles in terms of 𝑚 and 𝑛, so I can use the key fact that angles in a triangle sum to 180 degrees to form a second equation involving 𝑚 and 𝑛. If I add together the three angles in this triangle, I have nine 𝑚 minus two 𝑛 plus five 𝑚 plus two 𝑛 plus four 𝑚 is equal to 180. Now let’s try to simplify this equation. I have nine 𝑚 plus five 𝑚 plus four 𝑚, so that gives 18𝑚. Looking at the 𝑛s, I can see that I have minus two 𝑛 plus two 𝑛, so they cancel each other out directly. This just gives me then that 18𝑚 is equal to 180. Now in order to find the value of 𝑚, I can divide by 18. This tells me that 𝑚 is equal to 10.
So we’ve found the value of 𝑚, and if we recall that 𝑚 and 𝑛 are equal to each other, we’ve also found the value of 𝑛. They’re both equal to 10. And so we have an answer to the problem: both 𝑚 and 𝑛 are equal to 10.