1. **Problem Statement:** If a quadrilateral is a parallelogram, prove that its opposite sides are congruent and its diagonals bisect each other. 2. **Key Properties of a Parallelogram:** - Opposite sides are parallel. - Opposite sides are congruent. - Diagonals bisect each other. 3. **Proof for Opposite Sides Congruent:** Let the parallelogram be $ABCD$ with $AB \parallel DC$ and $AD \parallel BC$. By the properties of parallel lines and transversals, triangles $\triangle ABD$ and $\triangle CDB$ are congruent by the Side-Angle-Side (SAS) criterion: - $AB = DC$ (opposite sides) - $AD = BC$ (opposite sides) - Angles between these sides are equal because of parallel lines. Thus, $AB = DC$ and $AD = BC$. 4. **Proof for Diagonals Bisecting Each Other:** Let the diagonals $AC$ and $BD$ intersect at point $E$. Triangles $\triangle AEB$ and $\triangle CED$ are congruent by the Side-Angle-Side (SAS) criterion: - $AE = EC$ (to be proven) - $BE = ED$ (to be proven) - $AB = DC$ (already proven) Since these triangles are congruent, $AE = EC$ and $BE = ED$, meaning the diagonals bisect each other. **Final conclusion:** Opposite sides of a parallelogram are congruent and its diagonals bisect each other.