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)

Question

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)
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)

GPT-4o mini · Answer

To prove that opposite sides of a parallelogram are congruent, we'll present a two-column proof based on the properties of parallel lines and transversals.

Two-Column Proof

| Statement | Reason | |---------------------------------------------------|--------------------------------------------------------------| | 1. Let ABCD be a parallelogram with sides AB, BC, CD, DA. | Definition of a parallelogram. | | 2. Line a is parallel to line b, and line e is parallel to line f. | Given: a∥b and e∥f. | | 3. Line AB is parallel to line CD (AB || CD) and line AD is parallel to line BC (AD || BC). | Opposite sides of a parallelogram are parallel. | | 4. Angle ABE and angle CDF are alternate interior angles. | Corresponding properties of parallel lines with transversal AC. | | 5. Angle ABE ≅ Angle CDF | Alternate interior angles are congruent. | | 6. Angle DAB and angle BCD are alternate interior angles. | Corresponding properties of parallel lines with transversal BD. | | 7. Angle DAB ≅ Angle BCD | Alternate interior angles are congruent. | | 8. Triangle ABE is congruent to triangle CDF by the AA (Angle-Angle) criterion. | If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. | | 9. Therefore, corresponding sides AB and CD and AD and BC are congruent. | Corresponding parts of congruent triangles are congruent (CPCTC). | | 10. AB ≅ CD and AD ≅ BC. | Conclusion: Opposite sides of the parallelogram are congruent. |

Paragraph Proof

In a given parallelogram ABCD, we know that the opposite sides AB and CD are parallel (AB || CD) and AD is parallel to BC (AD || BC) due to the definition of a parallelogram. If we draw a transversal line AC, this creates alternate interior angles: angle ABE and angle CDF, which are congruent because they are formed by parallel lines and a transversal. Similarly, angle DAB and angle BCD are alternate interior angles that are also congruent. Because we have established that two angles of triangle ABE are congruent to two angles of triangle CDF, we can apply the AA (Angle-Angle) criterion of triangle similarity, confirming that triangle ABE is congruent to triangle CDF. By the property that corresponding parts of congruent triangles are congruent (CPCTC), we conclude that opposite sides AB and CD are congruent, as well as sides AD and BC. Thus, we have proved that the opposite sides of the parallelogram ABCD are congruent.