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 the properties of parallel lines and transversals. Below is a two-column proof that establishes this property.

Two-Column Proof

| Statements | Reasons | |------------------------------------------|-----------------------------------------| | 1. Let \( ABCD \) be a parallelogram where \( AB \parallel CD \) and \( AD \parallel BC \). | 1. Given. | | 2. \( AB \parallel CD \) and \( AD \parallel BC \) implies that \( AD \) and \( BC \) are both transversals. | 2. Properties of parallel lines. | | 3. Angle \( \angle DAB \) is congruent to angle \( \angle ABC \) (alternate interior angles). | 3. Alternate Interior Angles Theorem. | | 4. Angle \( \angle ADC \) is congruent to angle \( \angle BCD \) (alternate interior angles). | 4. Alternate Interior Angles Theorem. | | 5. Triangle \( \triangle DAB \) is similar to triangle \( \triangle ABC \) (by AA similarity). | 5. Angle-Angle (AA) Similarity Postulate. | | 6. Since \( \triangle DAB \sim \triangle ABC \), then the sides \( AB \) and \( CD \) are proportional to the sides \( AD \) and \( BC \). | 6. Corresponding sides of similar triangles are proportional. | | 7. In \( \triangle DAB \), let \( AD = BC \); thus, \( AB = CD \). | 7. Proportional sides of triangles. | | 8. Therefore, \( AB \cong CD \) and \( AD \cong BC \). | 8. Definition of congruence. |

Paragraph Proof

Let \( ABCD \) be a parallelogram where \( AB \parallel CD \) and \( AD \parallel BC \). By the properties of parallel lines, when a transversal crosses two parallel lines, alternate interior angles are congruent. Thus, in our parallelogram, \( \angle DAB \) is congruent to \( \angle ABC \) and \( \angle ADC \) is congruent to \( \angle BCD \). This creates two pairs of congruent angles. By the Angle-Angle (AA) similarity postulate, triangles \( \triangle DAB \) and \( \triangle ABC \) are similar. As a result, the corresponding sides of these triangles are proportional, leading us to conclude that \( AB \) is congruent to \( CD \) and \( AD \) is congruent to \( BC \). Therefore, we have proven that opposite sides of a parallelogram are congruent.