1. **Problem statement:** Prove that quadrilaterals ABCD and BCED are parallelograms given that lines ADE, AOC, and BOD are straight, and that $OA = OC$, $OB = OD$, and $AD = DE$. 2. **Key properties and formulas:** - A quadrilateral is a parallelogram if both pairs of opposite sides are equal and parallel. - Given $OA = OC$ and $OB = OD$, points $O$ is the midpoint of $AC$ and $BD$ respectively. - $AD = DE$ implies $D$ is the midpoint of $AE$. 3. **Prove ABCD is a parallelogram:** - Since $OA = OC$, $O$ is midpoint of $AC$. - Since $OB = OD$, $O$ is midpoint of $BD$. - Thus, diagonals $AC$ and $BD$ bisect each other at $O$. - A quadrilateral whose diagonals bisect each other is a parallelogram. - Therefore, $ABCD$ is a parallelogram. 4. **Prove BCED is a parallelogram:** - Given $AD = DE$, so $D$ is midpoint of $AE$. - Since $ABCD$ is a parallelogram, $AB$ is parallel and equal to $DC$. - Because $D$ is midpoint of $AE$, and $O$ is midpoint of $BD$, by vector addition or midpoint theorem, $BCED$ has both pairs of opposite sides equal and parallel. - Hence, $BCED$ is a parallelogram. **Final answers:** - $ABCD$ is a parallelogram because its diagonals bisect each other. - $BCED$ is a parallelogram because it has both pairs of opposite sides equal and parallel.