Question

given: $overline{ae}congoverline{ce};overline{de}congoverline{be}$
prove: $abcd$ is a parallelogram.
we have that $overline{ab}paralleloverline{dc}$.
by a similar argument used to prove that $\triangle aebcong\triangle ced$, we can show that $\trianglesquarecong\triangle ceb$ by $square$. so, $angle cadconganglesquare$ by cpctc. therefore, $adparallel bc$ by the converse of the $square$ theorem. since both pair of
alternate interior angles
alternate exterior angles
corresponding angles
quadrilateral $abcd$ is a

Explanation:

Step1: Prove triangle congruence

In $\triangle AED$ and $\triangle CEB$, we have $\overline{AE}\cong\overline{CE}$, $\overline{DE}\cong\overline{BE}$, and $\angle AED\cong\angle CEB$ (vertically - opposite angles). So, $\triangle AED\cong\triangle CEB$ by the Side - Angle - Side (SAS) congruence criterion.

Step2: Use CPCTC

By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), $\angle CAD\cong\angle ACB$.

Step3: Apply parallel - line theorem

Since $\angle CAD$ and $\angle ACB$ are alternate interior angles and they are congruent, by the converse of the alternate interior angles theorem, $AD\parallel BC$. We already know from proving $\triangle AEB\cong\triangle CED$ that $AB\parallel DC$. A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Answer:

We can show that $\triangle AED\cong\triangle CEB$ by SAS. So, $\angle CAD\cong\angle ACB$ by CPCTC. Therefore, $AD\parallel BC$ by the converse of the alternate interior angles theorem. Since both pair of opposite sides of quadrilateral $ABCD$ are parallel, $ABCD$ is a parallelogram.