In `Delta ABC BC bot AC` at D.E is a point on BC such that `angle BEA = x^(@)` If `angle EAC = 36^(@) and angle EBD = 40^(@)`, then the value of x is:

Options

A. `68^(@)`
B. `86^(@)`
C. `72^(@)`
D. `78^(@)`

Answer

To solve the problem, we will analyze the given triangle ABC and the points D and E on line segment BC. We will use the properties of angles in triangles to find the value of x.

### Step-by-Step Solution:

1. **Understand the Triangle Configuration**:
- We have triangle ABC where BC is perpendicular to AC. This means angle ABC = 90°.
- We are given angle EAC = 36° and angle EBD = 40°.

2. **Identify Angles in Triangle BDC**:
- In triangle BDC, we know:
- Angle EBD = 40° (given)
- Angle ABC = 90° (since BC is perpendicular to AC)
- Therefore, we can find angle BDC:
\[ \text{Angle BDC} = 180° - \text{Angle EBD} - \text{Angle ABC} = 180° - 40° - 90° = 50° \]

3. **Identify Angles in Triangle AEC**:
- Now we will analyze triangle AEC:
- We have angle EAC = 36° (given)
- We just calculated angle BDC = 50°, which is also angle AEC since they are the same angle in triangle AEC.
- We can find angle AEC:
\[ \text{Angle AEC} = 180° - \text{Angle EAC} - \text{Angle BDC} = 180° - 36° - 50° = 94° \]

4. **Find Angle BEA**:
- Now we know:
- Angle AEC = 94°
- Angle EBD = 40°
- We can find angle BEA (which is x):
\[ \text{Angle BEA} = \text{Angle AEC} - \text{Angle EBD} = 94° - 40° = 54° \]

5. **Conclusion**:
- Therefore, the value of x is:
\[ x = 54° \]

### Final Answer:
The value of x is **54°**.