### Using congruence to establish properties of parallelograms

#### Theorem

The opposite sides of a parallelogram are equal.

##### Proof

\(ABCD\) 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). |