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.(4 points)

Answer 1

To prove that opposite sides of a parallelogram are congruent, we can utilize a two-column proof format:

### Two-Column Proof

| Statement | Reason |
|-------------------------------------------------------|-----------------------------------------------------|
| 1. Let quadrilateral ABCD be a parallelogram where AB || CD and AD || BC. | Definition of a parallelogram. |
| 2. Line AB is parallel to line CD. | Given (AB || CD). |
| 3. Line AD is parallel to line BC. | Given (AD || BC). |
| 4. Angle A is equal to angle C (∠A = ∠C). | Alternate interior angles are congruent (AB || CD). |
| 5. Angle B is equal to angle D (∠B = ∠D). | Alternate interior angles are congruent (AD || BC). |
| 6. △ABD is similar to △CDB. | AA criterion for similarity (∠A ≅ ∠C and ∠B ≅ ∠D). |
| 7. AB/BD = AD/CD. | Corresponding sides of similar triangles are proportional. |
| 8. Since AD is a transversal, AD = BC (if we consider BD independent). | Properties of parallel lines and the definition of congruence. |
| 9. Thus, AB = CD and AD = BC. | If two sides are equal (from proportion), then they are congruent. |

### Conclusion:
The proof demonstrates that opposite sides of a parallelogram are congruent: AB is congruent to CD and AD is congruent to BC. Thus, all properties held true, confirming the congruence of opposite sides of a parallelogram.