In triangle ABC, the measure of angle C is four times the measure of angle A, and the measure of angle B is 18° more than the measure of angle C. Find the measure of each angle of the triangle.

Answer 1

Let's start by translate these statements into equations.

"the measure of angle C is four times the measure of angle A"

C = 4A

"the measure of angle B is 18 degrees more than the measure of angle C"

B = C + 18

That's all the information we are given about the angles, but we do at least know that these angles are part of a triangle, so we can also use facts about triangles. For any triangle, the angles should add to equal 180 degrees. So we can write a third equation:

180 = A + B + C

Now we have 3 equations and 3 variables to solve for. Let's solve using substitution. We can plug B = 4C + 18 into 180 = A + B + C, then simplify by combining like terms.

180 = A + C + 18 + C

180 = A + 2C + 18

Now this equation is in terms of two variables instead of three. If we can get that down to just one variable, we can solve for that variable and use it to find the other two. So let's write C in terms of A, and plug in C = 4A into our simplified equation.

180 = A + 2(4A) + 18, distribute the 2.

180 = A + 8A + 18, combine like terms.

180 = 9A + 18, subtract 18 from both sides

162 = 9A, divide by 9.

A = 18

When solving for A above, the goal is to get A by itself. So whatever is happening to A needs to be "undone" using the inverse operation, and whenever like terms are separated (two terms with A or two numeric terms) we combine them. Now that we know A = 18, we can plug that into our equation for C.

C = 4A

C = 4*18 = 72

And finally plug C = 72 into our equation for B.

B = C +18 = 72 + 18 = 90.

So our triangle ABC has angles that measure 18 degrees, 90 degrees, and 72 degrees. That means this a right triangle.