In `DeltaABC`, D and E are two mid points of sides AB and AC respec tively. If `/_BAC = 40^@` and `/_ABC = 65^@` then `/_CED `is

Options

A. `130^@`
B. `75^@`
C. `125^@`
D. `105^@`

Answer

To solve the problem, we need to find the measure of angle CED in triangle ABC, where D and E are the midpoints of sides AB and AC, respectively. We are given that angle BAC = 40° and angle ABC = 65°.

### Step-by-Step Solution:

1. **Identify the Angles in Triangle ABC**:
- We know that the sum of the angles in any triangle is 180°.
- Given:
- Angle BAC = 40°
- Angle ABC = 65°
- Let angle ACB be denoted as angle C.

\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180° \]
\[ 40° + 65° + \text{Angle C} = 180° \]

2. **Calculate Angle C**:
- Rearranging the equation to find angle C:
\[ \text{Angle C} = 180° - (40° + 65°) \]
\[ \text{Angle C} = 180° - 105° = 75° \]

3. **Understanding the Midpoints and Parallel Lines**:
- Since D and E are the midpoints of sides AB and AC, the line segment DE is parallel to side BC (by the Midpoint Theorem).
- This implies that quadrilateral BCEB is a trapezium.

4. **Using Properties of Trapezium**:
- In a trapezium, the sum of the adjacent angles is 180°.
- Therefore, we can write:
\[ \text{Angle BEC} + \text{Angle CED} = 180° \]

5. **Finding Angle BEC**:
- Since DE is parallel to BC, angle BEC is equal to angle ABC.
- Thus, angle BEC = 65°.

6. **Calculate Angle CED**:
- Now we can substitute angle BEC into the trapezium angle equation:
\[ 65° + \text{Angle CED} = 180° \]
\[ \text{Angle CED} = 180° - 65° = 115° \]

7. **Final Calculation**:
- Therefore, angle CED = 115°.

### Conclusion:
The measure of angle CED is 115°.