The difference between two angles of a triangle is 24°. The average of the same two angles is 54°. Which one of the following is the value of the greatest angle of the triangle?

Answer: Option D

Let a and b be the two angles in the question, with a > b. We are given that the difference between the angles is 24°.
⇒ a – b = 24
Since the average of the two angles is 54°, we have $$\frac{\text{a}+\text{b}}{2}$$  = 54
Solving for b in the first equation yields b = a – 24, and substituting this into the second equation yields,
$$\frac{\left\{\text{a}+\left(\text{a}-24\right)\right\}}{2}=54$$
2a − 24 = 54 × 2
2a − 24 = 108
2a = 108 + 24
2a = 132
a = 66
Also,
b = a − 24 = 66 − 24 = 42
Now, let c be the third angle of the triangle. Since the sum of the angles in the triangle is
180°, a + b + c = 180°
Putting the previous results into the equation yields 66 + 42 + c = 180°
Solving for c yields c = 72°
Hence, the greatest of the three angles a, b and c is c, which equal.