Question

Given: start overline, A, D, end overline, \parallel, start overline, B, C, end overline
AD
∥
BC
and start overline, A, B, end overline, \parallel, start overline, C, D, end overline, .
AB
∥
CD
.

Prove: angle, B, \cong, angle, D∠B≅∠D.

Answer 1

To prove that \(\angle B \cong \angle D\) given the conditions \(AD \parallel BC\) and \(AB \parallel CD\), we will use the properties of parallel lines and corresponding angles.

### Given:
1. \(AD \parallel BC\)
2. \(AB \parallel CD\)

### To Prove:
\(\angle B \cong \angle D\)

### Proof:
1. **Identify the setup**:
- We have two pairs of parallel lines:
- \(AD\) and \(BC\) are parallel.
- \(AB\) and \(CD\) are parallel.
- We will analyze the angles formed by these intersecting lines.

2. **Understanding the angles**:
- When transversal \(AB\) intersects the parallels \(AD\) and \(BC\), it creates angle \(B\) at the intersection of \(AB\) with \(BC\) and angle adjacent to it at \(D\) (let's denote the angle adjacent to \(D\) as \(\angle D\)) where \(CD\) intersects line \(AD\).

3. **Use the Corresponding Angles Theorem**:
- Based on the properties of parallel lines and transversals, we have:
- \(\angle B\) corresponds to \(\angle D\) because both pairs of lines are parallel.
- Specifically, since \(AB \parallel CD\) and \(AD \parallel BC\):
- \(\angle B\) (the angle on the same side of transversal \(AB\)) correlates with \(\angle D\) formed by the intersecting transversal \(CD\) with \(AD\).

4. **Conclusion**:
- As a result of the Corresponding Angles Postulate which states that if two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent:
\[
\angle B \cong \angle D
\]
Thus, we have proven that \(\angle B \cong \angle D\).

This concludes the proof.