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 paragraph proof or a two-column proof. Below, I will provide both formats for clarity.

Paragraph Proof

Given a parallelogram ABCD, where \(AB \parallel CD\) and \(AD \parallel BC\), we need to prove that \(AB \cong CD\) and \(AD \cong BC\).

1. Since \(AB \parallel CD\) and the transversal \(AD\) cuts these parallels, by the Alternate Interior Angles Theorem, we have \(\angle DAB \cong \angle BCD\).
2. Similarly, since \(AD \parallel BC\) and the transversal \(AB\) cuts these parallels, we have \(\angle ABC \cong \angle ADC\).
3. By the definition of a parallelogram, opposite angles are equal; thus, we have:
   • \(\angle DAB \cong \angle BCD\) (already established)
   • \(\angle ABC \cong \angle ADC\) (already established)
4. Triangles \(ABD\) and \(CDB\) are formed by the intersecting diagonals \(AC\) and \(BD\). We can see that:
   • \(AB\) is common to both triangles \(ABD\) and \(CDB\).
   • We have \(\angle DAB \cong \angle BCD\) and \(\angle ABC \cong \angle ADC\) from our previous statements.
5. By the Side-Angle-Side (SAS) Congruence Theorem, triangle \(ABD\) is congruent to triangle \(CDB\).
6. If two triangles are congruent, corresponding parts are congruent, therefore \(AD \cong BC\) and \(AB \cong CD\).
7. Hence, we conclude that in parallelogram ABCD, opposite sides \(AB\) and \(CD\) are congruent, as well as sides \(AD\) and \(BC\).

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. \(\angle DAB \cong \angle BCD\) | Alternate Interior Angles Theorem | | 4. \(\angle ABC \cong \angle ADC\) | Alternate Interior Angles Theorem | | 5. Triangles \(ABD\) and \(CDB\) are formed. | By drawing diagonal \(BD\) | | 6. \(AB\) is common to both triangles. | Common side of triangles | | 7. \(\angle DAB \cong \angle BCD\) (restated) | From step 3 | | 8. \(\angle ABC \cong \angle ADC\) (restated) | From step 4 | | 9. \(\triangle ABD \cong \triangle CDB\) | SAS Congruence Theorem | | 10. Therefore, \(AD \cong BC\) and \(AB \cong CD\) | Corresponding Parts of Congruent Triangles |

This proof demonstrates that the opposite sides of a parallelogram are congruent, using the properties of parallel lines and angles as well as the congruence of triangles.