# Proof of the theorem: Opposite sides of a parallelogram are congruent

 Given the following parallelogram on the right, prove that opposite sides are congruent or equal. Given: parallelogram ABCD

Prove: segment AB ≅ segment CD and segment BC ≅ AD

Since ABCD is a parallelogram, segment BC is parallel to segment AD according to the definition of a parallelogram.

By the same token, segment AB is parallel to segment DC according to the definition of a parallelogram.

Line AC is a transversal for segment BC and segment AD. This transversal creates alternate interior angles 1 and 4 which are equal as shown below. By the same token, Line AC is a transversal for segment AB and segment DC. This transversal creates alternate interior angles 2 and 3 which are equal. Moreover, segment AC ≅ segment AC by reflexive property.

Summary

∠1 ≅ ∠ 4

∠2 ≅ ∠ 3

segment AC ≅ segment AC

By ASA, triangle ABC ≅ triangle ADC

Therefore,segment AB ≅ segment CD and segment BC ≅ AD

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