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.