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?

A. 45°

B. 60°

C. 66°

D. 72°

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Solution(By Apex Team)

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 $\frac{a+b}{2}=54$ Solving for b in the first equation yields b = a – 24, and substituting this into the second equation yields, $\begin{array}{l}\left[\Large\frac{\mathrm{a}+(\mathrm{a}-24)}{2}\right]=54\\ 2\mathrm{a}-24=54\times2\\ 2\mathrm{a}-24=108\\ 2\mathrm{a}=108+24\\ 2\mathrm{a}=132\\ \mathrm{a}=66\\ \text{ Also, }\\ \mathrm{b}=\mathrm{a}-24=66-24=42\end{array}$ 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.