Question

In the diagram, a∥b and e∥f , forming a parallelogram. Write a proof that proves that opposite sides of the parallelogram are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements.(

Answer 1

Certainly! Here's a two-column proof that demonstrates that opposite sides of a parallelogram are congruent:

**Given:** A parallelogram ABCD where \( AB \parallel CD \) and \( AD \parallel BC \).

**To Prove:** \( AB \cong CD \) and \( AD \cong BC \).

| **Statements** | **Reasons** |
|--------------------------------------------------|----------------------------------------------------|
| 1. ABCD is a parallelogram. | 1. Given. |
| 2. \( AB \parallel CD \) and \( AD \parallel BC \). | 2. Definition of a parallelogram. |
| 3. \( ∠A \) and \( ∠D \) are alternate interior angles. | 3. Consecutive interior angles are supplementary, and corresponding angles are equal when lines are parallel. |
| 4. \( ∠A \cong ∠D \). | 4. Parallel lines cut by a transversal. |
| 5. \( ∠B \) and \( ∠C \) are also alternate interior angles. | 5. Same reason as above. |
| 6. \( ∠B \cong ∠C \). | 6. Parallel lines cut by a transversal. |
| 7. Triangles \( \triangle ABD \) and \( \triangle CDB \) are congruent. | 7. Angle-Angle (AA) criterion for triangle congruence (since \( \angle A \cong \angle D \) and \( \angle B \cong \angle C \)). |
| 8. \( AB \cong CD \). | 8. Corresponding parts of congruent triangles are congruent (CPCTC). |
| 9. Triangles \( \triangle ABD \) and \( \triangle CDB \) are congruent. | 9. Same reasoning as above. |
| 10. \( AD \cong BC \). | 10. Corresponding parts of congruent triangles are congruent (CPCTC). |

### Conclusion:
From statements 8 and 10, we conclude that opposite sides of the parallelogram \( ABCD \) are congruent: \( AB \cong CD \) and \( AD \cong BC \).

This proof utilizes the properties of parallel lines, congruences, and triangle characteristics to establish the congruence of opposite sides in a parallelogram.