In a triangle ABC, `angle B=angleC=78^(@)`. D & E are two points on side AB and AC such that `angle BCD=51^(@) and angle CBE=24^(@)`. Find angle `angle CDE=`?

Answer

To find the angle \( \angle CDE \) in triangle \( ABC \) with the given angles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Angles**:
- \( \angle B = \angle C = 78^\circ \)
- \( \angle BCD = 51^\circ \)
- \( \angle CBE = 24^\circ \)

2. **Calculate Angle \( A \)**:
- In triangle \( ABC \), the sum of angles is \( 180^\circ \).
\[ \angle A = 180^\circ - \angle B - \angle C = 180^\circ - 78^\circ - 78^\circ = 24^\circ \]

3. **Calculate Angle \( BDC \)**:
- In triangle \( BCD \):
\[ \angle BDC = 180^\circ - \angle B - \angle BCD = 180^\circ - 78^\circ - 51^\circ = 51^\circ \]

4. **Calculate Angle \( CBE \)**:
- In triangle \( CBE \):
\[ \angle CBE = 24^\circ \]
- Calculate angle \( CEB \):
\[ \angle CEB = 180^\circ - \angle C - \angle CBE = 180^\circ - 78^\circ - 24^\circ = 78^\circ \]

5. **Establish Relationships**:
- Since \( \angle BDC = 51^\circ \) and \( \angle CBE = 24^\circ \), we can find \( \angle CDE \).
- In triangle \( CDE \):
\[ \angle CDE = \angle CBE - \angle BDC = 24^\circ - 51^\circ = -27^\circ \text{ (not possible)} \]
- Instead, we need to calculate \( \angle BDE \) and \( \angle CDE \) correctly.

6. **Calculate Angle \( BDE \)**:
- Since \( \angle BDC = 51^\circ \) and \( \angle CBE = 24^\circ \):
\[ \angle BDE = 180^\circ - \angle CBE - \angle BDC = 180^\circ - 24^\circ - 51^\circ = 105^\circ \]

7. **Calculate Angle \( CDE \)**:
- Now, we can find \( \angle CDE \):
\[ \angle CDE = \angle BDE - \angle BDC = 105^\circ - 51^\circ = 54^\circ \]

### Final Answer:
\[ \angle CDE = 54^\circ \]