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 opposite sides of a parallelogram are congruent, we can use a two-column proof format. We will consider parallelogram \( ABCD \) where \( AB \parallel CD \) and \( AD \parallel BC \).

### Two-Column Proof

| Statement | Reason |
|--------------------------------------------------|------------------------------------------------|
| 1. Let \( ABCD \) be a parallelogram. | Given |
| 2. \( AB \parallel CD \) and \( AD \parallel BC \) | Definition of a parallelogram |
| 3. Draw diagonal \( AC \). | Construction |
| 4. Triangles \( \triangle ABC \) and \( \triangle CDA \) are formed. | Definition of triangles |
| 5. \( \angle ABC \cong \angle CDA \) | Alternate interior angles (due to \( AB \parallel CD \)) |
| 6. \( \angle ACB \cong \angle ACD \) | Alternate interior angles (due to \( AD \parallel BC \)) |
| 7. \( AC \) is common to both triangles. | Common side in triangles |
| 8. Therefore, by the Angle-Side-Angle (ASA) postulate, \( \triangle ABC \cong \triangle CDA \). | ASA Congruence Postulate |
| 9. Corresponding parts of congruent triangles are congruent, so \( AB \cong CD \). | CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
| 10. Also, by the same reasoning, \( AD \cong BC \). | CPCTC |

### Conclusion:
In conclusion, we have shown through a series of logical statements and reasons that opposite sides \( AB \) and \( CD \) are congruent, and consequently, that \( AD \) and \( BC \) are also congruent. This proves that in a parallelogram, opposite sides are congruent.