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Using congruence to establish properties of parallelograms

Theorem

The opposite sides of a parallelogram are equal.

Proof

Parallelogram ABCD with AD parallel to BC and AB parallel to DC and diagonal AC marked with a dotted line.

\(ABCD\) is a parallelogram. To prove that \(AB = CD \ \text{and} \ AD = BC\), join
the diagonal \(AC\) in the triangles \(ABC \ \text{and} \ CDA\).

\(\angle BAC\) = \(\angle DCA\) (alternate angles, \(AB \| DC\))
\(\angle BCA\) = \(\angle DAC\) (alternate angles, \(AD \| BC\))
\(AC\) = \(CA\) (common)
So \(\triangle ABC \equiv \triangle CDA\) (AAS)
Hence AB = CD and BC = AD (matching sides of congruent triangles).