Question

In a certain triangle, the angle of the greatest measure measures 75 degrees more than the angle of the least measure. If one-third the sum of the degree measures of the smallest and the largest angles is equal to the degree measure of the remaining angle, what is the degree measure of each angle of the triangle?

Answer 1

Let a<b<c

c=75+a  and (a+c)/3=b

using c from the first in the second you have:

b=(a+75+a)/3

b=(2a+75)/3

Since this is a triangle, a+b+c=180, using b and c from above we get:

a+(2a+75)/3+75+a=180  making the left side have a common denominator 3

(3a+2a+75+225+3a)/3=180  combine like terms on left side

(8a+300)/3=180 multiply both sides by 3

8a+300=540 subtract 300 from both sides

8a=240 divide both sides by 8

a=30°, since b=(2a+75)/3

b=45°, since c=75+a

c=105°

So the angles are 30°, 45°, and 105°