The Opposite Sides of a Parallelogram are Congruent

A parallelogram is a quadrilateral whose opposite sides are parallel. In this post, we show that asides from being parallel, they are also congruent. In the figure below, ABCD is a parallelogram; \overline{AB} is parallel to \overline{CD} and \overline{AD} is parallel to \overline{BC}.

A Parallelogram

 

To prove that the opposite sides of ABCD are congruent, we have to show that \overline{AD} \cong BC and \overline{AB} \cong CD.

Theorem: The opposite sides of a parallelogram are congruent.

Given: Parallelogram ABCD.

Proof: Draw \overline{BD}

parallelogram5

Notice that \overline{BD} serves as a transversal to the parallel line segments.

Clearly, \angle 1 \cong \angle 3 because they are alternate interior angles (A).

Also, BD \cong BD since any segment is congruent to itself (S).

Lastly, \angle 2 \cong \angle 4 because they are alternate interior angles (A)

Since the side is included by the two angles, by ASA Congruence, triangle ABC \cong CDB

Therefore, AB \cong CD and AD \cong BC since corresponding sides of congruent triangles are congruent. \blacksquare

So, the opposite sides of a parallelogram are congruent.

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03. March 2013 by Guillermo Bautista
Categories: Geometry, Grades 6-8, HS Math | Tags: , , | Leave a comment

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