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.
: parallelogram ABCD
: 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.
∠1 ≅ ∠ 4
∠2 ≅ ∠ 3
segment AC ≅ segment AC
By ASA, triangle ABC ≅ triangle ADC
Therefore,segment AB ≅ segment CD and segment BC ≅ AD
Dec 11, 17 08:45 AM
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