In `triangleABC "if" (AB)/(AC)= (BD)/(DC) and if angleB= 70^(@) and angleC = 50^(@) find angle BAD`

Answer

To solve the problem step by step, we will use the properties of triangles and the angle bisector theorem.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have triangle ABC.
- The ratio of sides is given as \( \frac{AB}{AC} = \frac{BD}{DC} \).
- Angles \( B \) and \( C \) are given as \( 70^\circ \) and \( 50^\circ \) respectively.

2. **Apply the Angle Bisector Theorem**:
- The condition \( \frac{AB}{AC} = \frac{BD}{DC} \) indicates that line segment \( AD \) bisects angle \( A \) (by the angle bisector theorem).

3. **Use the Angle Sum Property of a Triangle**:
- The sum of angles in a triangle is \( 180^\circ \).
- Therefore, we can write:
\[ \angle A + \angle B + \angle C = 180^\circ \]
- Substituting the known angles:
\[ \angle A + 70^\circ + 50^\circ = 180^\circ \]

4. **Solve for Angle A**:
- Rearranging the equation gives:
\[ \angle A = 180^\circ - 70^\circ - 50^\circ \]
- Simplifying this:
\[ \angle A = 180^\circ - 120^\circ = 60^\circ \]

5. **Find Angle BAD**:
- Since \( AD \) bisects angle \( A \), we have:
\[ \angle BAD = \angle CAD = \frac{1}{2} \angle A \]
- Substituting the value of \( \angle A \):
\[ \angle BAD = \frac{1}{2} \times 60^\circ = 30^\circ \]

6. **Conclusion**:
- Therefore, the measure of angle \( BAD \) is \( 30^\circ \).

### Final Answer:
\[ \angle BAD = 30^\circ \]