To prove that opposite sides of a parallelogram are congruent, we can use a two-column proof. Let's denote the vertices of the parallelogram as \( ABCD \) with sides \( AB \parallel CD \) and \( AD \parallel BC \).
Two-Column Proof
| Statements | Reasons | |---------------------------------------------------|-------------------------------------------------| | 1. Let \( AB \parallel CD \) and \( AD \parallel BC \) in parallelogram \( ABCD \). | 1. Given. | | 2. Draw diagonal \( AC \). | 2. A diagonal can be drawn in a parallelogram. | | 3. Triangles \( \triangle ABC \) and \( \triangle CDA \) are formed by the diagonal \( AC \). | 3. Definition of triangles. | | 4. Angles \( \angle ABC \) and \( \angle CDA \) are alternate interior angles. | 4. Parallel lines cut by a transversal form alternate interior angles. | | 5. \( \angle ABC \cong \angle CDA \) (alternate interior angles). | 5. Properties of parallel lines. | | 6. Angles \( \angle ACB \) and \( \angle DCA \) are also alternate interior angles. | 6. Parallel lines cut by a transversal form alternate interior angles. | | 7. \( \angle ACB \cong \angle DCA \) (alternate interior angles). | 7. Properties of parallel lines. | | 8. \( AC \) is common to both triangles \( \triangle ABC \) and \( \triangle CDA \). | 8. Common side of triangles. | | 9. By the Angle-Side-Angle (ASA) postulate, \( \triangle ABC \cong \triangle CDA \). | 9. ASA Postulate (two angles and the included side are congruent). | | 10. Therefore, \( AB \cong CD \) and \( AD \cong BC \). | 10. Corresponding parts of congruent triangles are congruent (CPCTC). |
Conclusion
Thus, we have proven that the opposite sides of the parallelogram \( ABCD \) are congruent, i.e., \( AB \cong CD \) and \( AD \cong BC \).