The internal angles of a triangle are in A.P. . If the smallest angle is `45^(@)`, find the remaining angles.

Answer

To solve the problem, we need to find the internal angles of a triangle where the angles are in Arithmetic Progression (A.P.) and the smallest angle is given as \(45^\circ\).

### Step-by-Step Solution:

1. **Understanding the Angles in A.P.**:
- Let's denote the three angles of the triangle as \(A\), \(A + D\), and \(A + 2D\), where \(A\) is the smallest angle and \(D\) is the common difference.
- Given that the smallest angle \(A = 45^\circ\).

2. **Setting Up the Equation**:
- The sum of the angles in a triangle is \(180^\circ\). Therefore, we can write the equation:
\[ A + (A + D) + (A + 2D) = 180^\circ \]
- Substituting \(A = 45^\circ\) into the equation:
\[ 45 + (45 + D) + (45 + 2D) = 180 \]

3. **Simplifying the Equation**:
- Combine the terms:
\[ 45 + 45 + D + 45 + 2D = 180 \]
\[ 135 + 3D = 180 \]

4. **Solving for D**:
- Subtract \(135\) from both sides:
\[ 3D = 180 - 135 \]
\[ 3D = 45 \]
- Divide both sides by \(3\):
\[ D = 15 \]

5. **Finding the Remaining Angles**:
- Now that we have \(D\), we can find the other angles:
- Second angle: \(A + D = 45 + 15 = 60^\circ\)
- Third angle: \(A + 2D = 45 + 2(15) = 45 + 30 = 75^\circ\)

6. **Final Angles**:
- The angles of the triangle are:
\[ 45^\circ, 60^\circ, 75^\circ \]

### Summary of the Solution:
The internal angles of the triangle are \(45^\circ\), \(60^\circ\), and \(75^\circ\).