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.

Question

The parallelogram created by a||b and e||f is named WYZX and it is split into two triangles by a diagonal going from point Y to point X. Creating triangle YWX (Y = 1, W = 2, X = 3) and triangle XZY (X = 4, Z = 5, Y = 6.

Answer 1

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

Given: \(a \parallel b\) and \(e \parallel f\), forming parallelogram WYZX.

To Prove: \(WY \cong XZ\) and \(WX \cong YZ\).

Two-Column Proof

| Statements | Reasons | |----------------------------------------------------|--------------------------------------------------| | 1. \(a \parallel b\) and \(e \parallel f\) | 1. Given | | 2. Diagonal \(YX\) divides parallelogram WYZX into triangles \(YWX\) and \(YZX\) | 2. Definition of a diagonal | | 3. Angles \(YWX\) and \(YZX\) are alternate interior angles (formed by transversal line \(WX\)) | 3. Alternate interior angle theorem | | 4. \(\angle YWX \cong \angle YZX\) | 4. Alternate interior angles are congruent | | 5. Angles \(WYB\) and \(ZXY\) are alternate interior angles (formed by transversal line \(YZ\)) | 5. Alternate interior angle theorem | | 6. \(\angle WYZ \cong \angle ZYX\) | 6. Alternate interior angles are congruent | | 7. Triangle \(YWX \cong \triangle YZX\) | 7. Angle-Angle (AA) similarity criterion | | 8. \(WY \cong XZ\) | 8. Corresponding parts of congruent triangles are congruent (CPCTC) | | 9. \(WX \cong YZ\) | 9. Corresponding parts of congruent triangles are congruent (CPCTC) |

Conclusion

From the above proof, we conclude that in the parallelogram WYZX, the opposite sides \(WY\) is congruent to \(XZ\) and \(WX\) is congruent to \(YZ\). Thus, the opposite sides of the parallelogram are confirmed to be congruent.