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

Explanation:-

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^o.
=> a – b = 24.
Since the average of the two angles is 54^o, 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,
$$\left[ {\frac{{\left\{ {{\text{a}} + \left( {{\text{a}} - 24} \right)} \right\}}}{2}} \right] = 54$$
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^o, 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