### 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°

## Show Answer

### Answer-D

Solution-

__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.

Thank you sir