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?

Options

A. `45^@`
B. `60^@`
C. `66^@`
D. `72^@`

Answer

To solve the problem step by step, we will use the information given about the angles of the triangle.

### Step 1: Define the angles
Let the two angles be \( a \) and \( b \). According to the problem, the difference between these two angles is given as:
\[ a - b = 24^\circ \quad \text{(1)} \]

### Step 2: Use the average of the angles
The average of the two angles is given as \( 54^\circ \). The formula for the average of two numbers is:
\[ \text{Average} = \frac{a + b}{2} \]
Setting this equal to \( 54^\circ \):
\[ \frac{a + b}{2} = 54^\circ \]
Multiplying both sides by 2 gives:
\[ a + b = 108^\circ \quad \text{(2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( a - b = 24^\circ \) (from step 1)
2. \( a + b = 108^\circ \) (from step 2)

We can add these two equations to eliminate \( b \):
\[ (a - b) + (a + b) = 24^\circ + 108^\circ \]
This simplifies to:
\[ 2a = 132^\circ \]
Dividing both sides by 2 gives:
\[ a = 66^\circ \quad \text{(3)} \]

### Step 4: Find the value of \( b \)
Now we can substitute the value of \( a \) back into equation (1) to find \( b \):
\[ 66^\circ - b = 24^\circ \]
Rearranging gives:
\[ b = 66^\circ - 24^\circ = 42^\circ \quad \text{(4)} \]

### Step 5: Find the third angle \( c \)
The sum of the angles in a triangle is \( 180^\circ \). Therefore, we can find the third angle \( c \):
\[ a + b + c = 180^\circ \]
Substituting the values of \( a \) and \( b \):
\[ 66^\circ + 42^\circ + c = 180^\circ \]
This simplifies to:
\[ 108^\circ + c = 180^\circ \]
Subtracting \( 108^\circ \) from both sides gives:
\[ c = 180^\circ - 108^\circ = 72^\circ \quad \text{(5)} \]

### Step 6: Identify the greatest angle
Now we have the values of the three angles:
- \( a = 66^\circ \)
- \( b = 42^\circ \)
- \( c = 72^\circ \)

The greatest angle among these is:
\[ c = 72^\circ \]

### Final Answer
The value of the greatest angle of the triangle is:
\[ \boxed{72^\circ} \]