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, $latex ABCD$ is a parallelogram; $latex \overline{AB}$ is parallel to $latex \overline{CD}$ and $latex \overline{AD}$ is parallel to $latex \overline{BC}$.

A Parallelogram

 

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

Theorem: The opposite sides of a parallelogram are congruent.

Given: Parallelogram $latex ABCD$.

Proof: Draw $latex \overline{BD}$

parallelogram5

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

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

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

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

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

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

So, the opposite sides of a parallelogram are congruent.

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