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.(4 points)

Answer 1

Certainly! Let's work through a paragraph proof to demonstrate that opposite sides of a parallelogram are congruent.

Proof:

To prove that the opposite sides of a parallelogram are congruent, we begin by considering the properties of a parallelogram. Let \( ABCD \) be a parallelogram where \( AB \parallel CD \) and \( AD \parallel BC \).

1. Since \( AB \parallel CD \) and \( AD \parallel BC \), we can use the Alternate Interior Angles Theorem. This theorem states that if two lines are parallel, the angles formed by a transversal line intersecting these lines are congruent.

2. In our parallelogram, line \( AD \) acts as a transversal to lines \( AB \) and \( CD \). Therefore, the angles \( \angle DAB \) and \( \angle ABC \) are congruent (as they are alternate interior angles).

3. Similarly, \( BC \) acts as a transversal to lines \( AD \) and \( AB \), hence \( \angle ABC \) and \( \angle BCD \) are also congruent.

4. Now consider triangles \( ABD \) and \( CDB \). These triangles share side \( BD \) as a common side, and we have established that \( \angle DAB \cong \angle ABC \) and \( \angle ABD \cong \angle CDB \) (by the property of alternate interior angles).

5. By the Angle-Side-Angle (ASA) postulate, we can conclude that triangle \( ABD \cong \triangle CDB \).

6. Because the triangles are congruent, corresponding sides of congruent triangles must also be congruent, thus we find \( AB \cong CD \) and \( AD \cong BC \).

7. This shows that opposite sides of parallelogram \( ABCD \) are congruent.

Therefore, we have proven that in a parallelogram, the opposite sides are congruent using properties of parallel lines and triangles.

This completes our proof.