Question

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is four times the measure of the first angle. The third angle is 12 more than the second. Let x, y, and z represent the measures of the first, second, and third angles, respectively. Find the measures of the three angles.

Answer 1

{36, 66, 78}

Explanation:

Let the measures of the three angles be f, s and t, for first, second and third. Then s + t = 4f, t = 12 + s, and f + s + t = 180.

Subbing 12 + s for t in the first equation, we get s + 12 + s = 4f.

Subbing the same in the third equation, we get f + s + 12 + s = 180

This results in two equations in two unknowns (f and s):

2s - 4f = -12

2s + f = 168.

Let's eliminate s by subtracting the first of these two equations from the second. Doing so yields 5f = 180. Then the first angle, f, or x, is 36.

Then, from 2s + f = 168, we get 2s + 36 = 168, or 2s = 132. Thus, the second angle is s = y = 66.

The sum of the three angles must be 180. Thus, 36 + 66 + t = 180, or

102 + t = 180, or t = z = 78.

The three angles are

{36, 66, 78}.