The difference between two supplementary angles is `72^@` Find the angles.

Answer

To solve the problem of finding two supplementary angles where the difference between them is 72 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Supplementary Angles**:
Supplementary angles are two angles whose sum is 180 degrees. Let's denote the two angles as \( x \) and \( y \).

2. **Setting Up the Equations**:
From the problem, we know:
- The sum of the angles:
\[ x + y = 180 \quad \text{(Equation 1)} \]
- The difference between the angles:
\[ x - y = 72 \quad \text{(Equation 2)} \]

3. **Adding the Two Equations**:
To eliminate \( y \), we can add Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 180 + 72 \]
This simplifies to:
\[ 2x = 252 \]

4. **Solving for \( x \)**:
Now, divide both sides by 2 to find \( x \):
\[ x = \frac{252}{2} = 126 \]

5. **Finding \( y \)**:
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):
\[ 126 + y = 180 \]
Rearranging gives:
\[ y = 180 - 126 = 54 \]

6. **Conclusion**:
The two supplementary angles are:
\[ x = 126^\circ \quad \text{and} \quad y = 54^\circ \]

### Verification:
To verify, we can check:
- The sum:
\[ 126 + 54 = 180 \quad \text{(Correct)} \]
- The difference:
\[ 126 - 54 = 72 \quad \text{(Correct)} \]

Thus, the angles are \( 126^\circ \) and \( 54^\circ \).