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

Paragraph Proof:

To prove that the opposite sides of a parallelogram are congruent, let's consider parallelogram ABCD, where sides AB and CD are opposite sides, and sides BC and AD are the other pair of opposite sides.

1. Since lines a and b are parallel (a∥b) and lines e and f are also parallel (e∥f), by the definition of a parallelogram, we conclude that AB is parallel to CD and BC is parallel to AD. (Definition of a parallelogram)

2. Because AB is parallel to CD, and they are cut by the transversal BC, we can use the Alternate Interior Angles Theorem, which tells us that the interior angles ∠ABC and ∠BCD are equal. (Alternate Interior Angles Theorem)

3. Similarly, because BC is parallel to AD and they are cut by transversal AB, the angles ∠CAB and ∠ADC are also equal by the Alternate Interior Angles Theorem. (Alternate Interior Angles Theorem)

4. Now, we will use the fact that triangles ABC and CDA are formed by the diagonals AC and BD intersecting at point O. We can state that triangles ABC and CDA are congruent by the Angle-Angle (AA) criterion for triangle congruence, as they have two angles equal (∠ABC = ∠BCD and ∠CAB = ∠ADC). (AA Criterion for Triangle Congruence)

5. By the property of congruent triangles, if triangle ABC is congruent to triangle CDA, then the corresponding sides are congruent. Therefore, we have AB = CD and BC = AD. (Corresponding Parts of Congruent Triangles are Congruent Postulate)

Thus, we have proven that the opposite sides of a parallelogram are congruent, as desired.

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Two-Column Proof:

| Statement | Reason | |---------------------------------------------|----------------------------------------------------------| | 1. ABCD is a parallelogram | Definition of a parallelogram | | 2. AB ∥ CD and BC ∥ AD | Properties of parallelograms | | 3. ∠ABC = ∠BCD | Alternate Interior Angles Theorem | | 4. ∠CAB = ∠ADC | Alternate Interior Angles Theorem | | 5. ΔABC ≅ ΔCDA | Angle-Angle (AA) Criterion for Triangle Congruence | | 6. AB = CD and BC = AD | Corresponding Parts of Congruent Triangles are Congruent (CPCTC) |

This two-column proof shows the statements and the reasons for every step made to conclude that opposite sides of a parallelogram are congruent.