In a parallelogram ABCD, the bisector of `anlgeA` also bisects BC at E, find the AD.

Options

A. AB
B. 2AB
C. 3AB
D. `1/2`AB

Answer

To solve the problem, we need to find the length of side AD in parallelogram ABCD, given that the angle bisector of angle A also bisects side BC at point E.

### Step-by-Step Solution:

1. **Understanding the Properties of a Parallelogram**:
In a parallelogram, opposite sides are equal and parallel. Therefore, we have:
- AB = CD
- AD = BC

2. **Identifying the Angles**:
Let angle A = ∠A and angle B = ∠B. In a parallelogram, we know that:
- ∠A + ∠B = 180° (consecutive angles are supplementary).

3. **Using the Angle Bisector**:
Since AE is the angle bisector of ∠A, we can say:
- ∠BAE = ∠EAD = ½ ∠A

4. **Applying the Angle Bisector Theorem**:
The angle bisector theorem states that the ratio of the lengths of the two segments created by the angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides. Therefore, we have:
\[ \frac{BE}{EC} = \frac{AB}{AD} \]

5. **Setting Up the Equation**:
Since AB = CD and AD = BC, we can denote:
- AB = x
- AD = y
Thus, we can rewrite the ratio:
\[ \frac{BE}{EC} = \frac{x}{y} \]

6. **Using the fact that E bisects BC**:
Since E bisects BC, we have:
\[ BE = EC \]
Let BE = EC = k. Thus, BC = BE + EC = k + k = 2k.

7. **Substituting into the Ratio**:
Since BC = AD, we can substitute:
\[ \frac{k}{k} = \frac{x}{y} \implies 1 = \frac{x}{y} \]
This implies that:
\[ x = y \]

8. **Conclusion**:
Therefore, we find that:
\[ AD = AB \]
Hence, in parallelogram ABCD, since the angle bisector of angle A bisects BC at E, we conclude that AD is equal to AB.

### Final Answer:
AD = AB

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