Quadrilateral with Congruent Opposite Sides are Parallelogram

This is the first of a series of post about the conditions of a quadrilateral to be a parallelogram.  In this post, I will be discussing the proof that if the opposite sides of a quadrilateral are congruent, then it is a parallelogram.



Quadrilateral ABCD

\overline{AB} \cong \overline{CD}

\overline {BC} \cong \overline{AD} 


Draw diagonal \overline{BD}.


Now, \overline{AB} \cong \overline{CD} (S) and \overline {BC} \cong \overline{AD} (S).

Also, a segment is congruent to itself (Relfexive Property), so \overline{BD} \cong \overline{BD} (S).

Therefore, by the SSS Triangle Congruence, \triangle ABD \cong \triangle BDA.

Since corresponding angles of congruent triangles are congruent,

\angle ADB \cong \angle CBD and \angle ABD \cong \angle CDB.

But these pairs of angles are corresponding angles, so by the Parallel Line Postulate,

AB \parallel DC and AD \parallel BD.

So, quadrilateral ABCD is a parallelogram.

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