The angles of a triangle are in A.P. and tangent of smallest angle is 1, then the angles are `45^(@), 60^(@), 75^(@)`. T or F ?

Answer

To determine if the statement "The angles of a triangle are 45°, 60°, and 75°" is true or false, given that the angles are in Arithmetic Progression (A.P.) and the tangent of the smallest angle is 1, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that the angles of a triangle sum up to 180°.
- The angles are in A.P., which means they can be represented as \(a - d\), \(a\), and \(a + d\) for some angle \(a\) and common difference \(d\).
- The smallest angle is given to have a tangent of 1.

2. **Finding the Smallest Angle**:
- The tangent of the smallest angle being 1 implies that this angle is 45° (since \(\tan 45° = 1\)).
- Therefore, we can set the smallest angle \(A = 45°\).

3. **Setting Up the Angles**:
- Let the angles be \(A\), \(B\), and \(C\) such that:
- \(A = 45°\)
- \(B = 45° + d\)
- \(C = 45° + 2d\)

4. **Using the Triangle Angle Sum Property**:
- The sum of the angles in a triangle is 180°:
\[ A + B + C = 180° \]
- Substituting the values we have:
\[ 45° + (45° + d) + (45° + 2d) = 180° \]
- Simplifying this:
\[ 135° + 3d = 180° \]
\[ 3d = 180° - 135° \]
\[ 3d = 45° \]
\[ d = 15° \]

5. **Calculating the Angles**:
- Now we can find the angles:
- \(A = 45°\)
- \(B = 45° + 15° = 60°\)
- \(C = 45° + 2(15°) = 75°\)

6. **Conclusion**:
- The angles of the triangle are \(45°\), \(60°\), and \(75°\).
- Therefore, the statement "The angles of a triangle are 45°, 60°, and 75°" is **True**.