An angle of a triangle mesures `68^(@)` and the other two angles differ by `16^(@)` . Find the angles.

Answer

To solve the problem, we need to find the angles of a triangle given that one angle measures \(68^\circ\) and the other two angles differ by \(16^\circ\).

### Step-by-Step Solution:

1. **Identify the Angles**:
Let the angles of the triangle be \(A\), \(B\), and \(C\). We know that:
- \(A = 68^\circ\)
- The difference between angles \(B\) and \(C\) is \(16^\circ\). We can express this as:
\[ B - C = 16^\circ \]

2. **Express \(C\) in terms of \(B\)**:
From the equation \(B - C = 16^\circ\), we can express \(C\) as:
\[ C = B - 16^\circ \]

3. **Use the Angle Sum Property**:
The sum of the angles in a triangle is always \(180^\circ\). Therefore, we can write:
\[ A + B + C = 180^\circ \]
Substituting \(A\) and \(C\) into this equation gives:
\[ 68^\circ + B + (B - 16^\circ) = 180^\circ \]

4. **Simplify the Equation**:
Combine like terms:
\[ 68^\circ + B + B - 16^\circ = 180^\circ \]
This simplifies to:
\[ 2B + 52^\circ = 180^\circ \]

5. **Isolate \(B\)**:
Subtract \(52^\circ\) from both sides:
\[ 2B = 180^\circ - 52^\circ \]
\[ 2B = 128^\circ \]
Now, divide by \(2\):
\[ B = 64^\circ \]

6. **Find \(C\)**:
Now that we have \(B\), we can find \(C\) using the equation \(C = B - 16^\circ\):
\[ C = 64^\circ - 16^\circ = 48^\circ \]

7. **Summarize the Angles**:
Now we have all the angles of the triangle:
- \(A = 68^\circ\)
- \(B = 64^\circ\)
- \(C = 48^\circ\)

### Final Answer:
The angles of the triangle are:
- \(A = 68^\circ\)
- \(B = 64^\circ\)
- \(C = 48^\circ\)