A circle is the circumcircle for the triangle DEF and the incircle for the triangle ABC. If the angles of triangle DEF are `76^@, 56^@ and 48^@`, find the values of internal angles of `triangleABC`.

Options

A. 32,46,92
B. 28,68,84
C. 38,48,74
D. data insufficient

Answer

To solve the problem, we need to find the internal angles of triangle ABC given that triangle DEF has angles of 76°, 56°, and 48°. Since the circle is the circumcircle of triangle DEF and the incircle of triangle ABC, we can use the properties of angles in triangles.

### Step-by-Step Solution:

1. **Understand the Angles of Triangle DEF**:
- The angles of triangle DEF are given as:
- Angle D = 76°
- Angle E = 56°
- Angle F = 48°

2. **Calculate the Sum of Angles in Triangle DEF**:
- The sum of the angles in any triangle is 180°:
\[ 76° + 56° + 48° = 180° \]
- This confirms that the angles provided are correct.

3. **Use the Property of Circumcircle and Incircle**:
- The angles of triangle ABC can be found using the relationship between the angles of the triangle DEF and the angles of triangle ABC.
- The angles of triangle ABC are related to the angles of triangle DEF as follows:
- Angle A = 90° - Angle D
- Angle B = 90° - Angle E
- Angle C = 90° - Angle F

4. **Calculate the Angles of Triangle ABC**:
- Calculate Angle A:
\[ A = 90° - 76° = 14° \]
- Calculate Angle B:
\[ B = 90° - 56° = 34° \]
- Calculate Angle C:
\[ C = 90° - 48° = 42° \]

5. **Final Angles of Triangle ABC**:
- The internal angles of triangle ABC are:
- Angle A = 14°
- Angle B = 34°
- Angle C = 42°

### Summary of Internal Angles of Triangle ABC:
- Angle A = 14°
- Angle B = 34°
- Angle C = 42°