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The problem asks you to prove that \(\angle B \cong \angle D\), given the parallel lines \(BC \parallel AD\) and \(AB \parallel CD\).

Here’s how we can complete the proof:

1. **Given**: 
   - \( BC \parallel AD \)
   - \( AB \parallel CD \)
   This information is already given in the proof table.

2. **Reasoning through parallel lines**:
   - By the properties of parallel lines cut by a transversal, corresponding angles are congruent.
   - The angles \(\angle BAC \cong \angle ACD\) because they are corresponding angles formed by the transversal \(AC\).

3. **Vertical angles**:
   - Now, observe that \(\angle B \) and \(\angle D\) are vertical angles.
   - Vertical angles are always congruent.

Thus, the steps in the proof are:
1. **Given**: \(BC \parallel AD\) and \(AB \parallel CD\).
2. **Statement**: \(\angle BAC \cong \angle ACD\), reason: Parallel lines cut by a transversal.
3. **Statement**: \(\angle B \cong \angle D\), reason: Vertical angles are congruent.

This completes the proof.

Would you like to go over any specific step in more detail or have further questions?

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Here are 5 related questions to expand your understanding:
1. What other pairs of angles could be proven congruent using the same reasoning in this diagram?
2. How do corresponding angles work when parallel lines are cut by multiple transversals?
3. Why are vertical angles always congruent?
4. How could you use the properties of parallel lines to prove congruence in different quadrilateral shapes?
5. Could this proof be extended to show congruence of other angle pairs in the diagram?

**Tip**: Whenever parallel lines are involved, always check for corresponding, alternate interior, or vertical angles

Given BC || AD and AB || CD, prove that angle B is congruent to angle D.

This problem involves proving the congruence of angles in a geometric setup with parallel lines. Learn how to apply the properties of parallel lines and vertical angles to prove that angle B is congruent to angle D. Suitable for students in Grades 9-10.

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