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

To prove that the opposite sides of a parallelogram are congruent, we can use a two-column proof format.

### Two-Column Proof

| Statement | Reason |
|-----------|--------|
| 1. Let parallelogram be ABCD with sides AB, BC, CD, and DA. | Definition of a parallelogram. |
| 2. Line segments AB and CD are opposite sides of the parallelogram. | Definition of the sides of a parallelogram. |
| 3. Line segments AD and BC are opposite sides of the parallelogram. | Definition of the sides of a parallelogram. |
| 4. Since AB is parallel to CD and AD is parallel to BC, by the properties of parallel lines, corresponding angles are equal. | Corresponding Angles Postulate. |
| 5. Let the transversal lines AD and BC intersect lines AB and CD. | Definition of transversal lines. |
| 6. ∠ABD ≅ ∠CDB and ∠Bad ≅ ∠DAB | Since AB || CD and AD || BC, corresponding angles are congruent. |
| 7. Triangles ABD and CDB are formed. | Construction, noting the common side BD. |
| 8. By the Angle-Side-Angle (ASA) postulate, triangles ABD and CDB are congruent. | If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. |
| 9. Therefore, corresponding sides of triangles ABD and CDB are congruent. | CPCTC (Corresponding Parts of Congruent Triangles are Congruent). |
| 10. Thus, AB ≅ CD and AD ≅ BC. | From the previous step, we conclude the opposite sides are congruent. |

### Conclusion
This proof demonstrates that the opposite sides of a parallelogram are congruent by using properties of parallel lines, angles, and triangle congruence.