Use the image to answer the question.

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)

Short Answer Rubric (4 points)

Answer 1

Two-Column Proof:

| Statement | Reason | |-------------------------------------------------------------------|-----------------------------------------------------------| | 1. Let ABCD be a parallelogram where AB || CD and AD || BC. | Given: Definition of a parallelogram | | 2. AB || CD and AD || BC | Given: Properties of parallel lines | | 3. Angles ∠A and ∠B are alternate interior angles. | Alternate Interior Angles Theorem (since AB || CD) | | 4. Angle ∠A ≈ Angle ∠B | If parallel lines are cut by a transversal, then the alternate interior angles are congruent. | | 5. Angles ∠D and ∠C are alternate interior angles. | Alternate Interior Angles Theorem (since AD || BC) | | 6. Angle ∠D ≈ Angle ∠C | Same as step 4, alternate interior angles are congruent. | | 7. Triangle ABD ≈ Triangle CDB | AA (Angle-Angle) similarity criterion (congruent angles) | | 8. AB ≈ CD and AD ≈ BC | Corresponding sides of congruent triangles are congruent. | | 9. Therefore, opposite sides AB and CD are congruent, and sides AD and BC are congruent. | Definition of congruent sides in a parallelogram |

Summary:

In this proof, we established that in a parallelogram, opposite sides are congruent by demonstrating that the triangles formed by connecting the diagonals show angle congruencies, leading to the conclusion that the sides themselves must also be congruent.