Use the image to answer the 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)

Question

Use the image to answer the 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)

GPT-4o mini · Answer

To prove that opposite sides of a parallelogram are congruent, you can use the following two-column proof format:

Two-Column Proof

| Statement | Reason | |-------------------------------------------------------|------------------------------------------| | 1. In parallelogram ABCD, let AB || CD and AD || BC. | Definition of a parallelogram | | 2. Line AB is parallel to Line CD. | Given (a ∥ b) | | 3. Line AD is parallel to Line BC. | Given (e ∥ f) | | 4. Angle A is congruent to Angle C (∠A ≅ ∠C). | Alternate interior angles are congruent (from parallel lines AB and CD) | | 5. Angle B is congruent to Angle D (∠B ≅ ∠D). | Alternate interior angles are congruent (from parallel lines AD and BC) | | 6. Triangle ABD is congruent to triangle CDB. | Angle-Angle (AA) similarity | | 7. Therefore, AB ≅ CD. | Corresponding parts of congruent triangles are congruent (CPCTC) | | 8. Triangle ABC is congruent to triangle ADC. | Angle-Angle (AA) similarity | | 9. Therefore, BC ≅ AD. | Corresponding parts of congruent triangles are congruent (CPCTC) |

Conclusion:

We've established that opposite sides of parallelogram ABCD, namely AB and CD, as well as AD and BC, are congruent, proving that opposite sides of a parallelogram are congruent.

Paragraph Proof

In parallelogram ABCD, by definition, opposite sides AB and CD are parallel, while AD and BC are also parallel. Since AB is parallel to CD, the alternate interior angles created by any transversal (e.g., line AC) are congruent; hence, angle A is congruent to angle C (∠A ≅ ∠C). Similarly, angle B is congruent to angle D (∠B ≅ ∠D) because AD is parallel to BC. By the Angle-Angle (AA) similarity criterion, triangle ABD is congruent to triangle CDB. By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we conclude that AB ≅ CD. A similar argument holds for triangles ABC and ADC, leading to the conclusion that BC ≅ AD. Therefore, we have proved that opposite sides of a parallelogram are congruent.