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 $\triangle cong\triangle ceb$ by
so, $angle cadcongangle $ by cpctc. therefore,
$adparallel bc$ by the converse of the
theorem. since both pair of
opposite sides are parallel, quadrilateral abcd is a
parallelogram.

Explanation:

Step1: Identify congruent triangles

In $\triangle AED$ and $\triangle CEB$, we have $AE\cong CE$, $DE\cong BE$, 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 converse of alternate - interior angles theorem

Since $\angle CAD\cong\angle ACB$, by the converse of the alternate - interior angles theorem, $AD\parallel BC$. Given $AB\parallel DC$, and now $AD\parallel BC$, by the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel), $ABCD$ is a parallelogram.

Answer:

In the blanks: $\triangle AED$, SAS, $\angle ACB$, alternate - interior angles.