Question

the figure below shows a quadrilateral abcd. sides ab and dc are congruent and parallel: a student wrote the following sentences to prove that quadrilateral abcd is a parallelogram: side ab is parallel to side dc, so the alternate interior angles, angle abd and angle cdb, are congruent. side ab is equal to side dc, and db is the side common to triangles abd and cdb. therefore, the triangles abd and cdb are congruent by ________. by cpctc, angles dbc and adb are congruent and sides ad and bc are congruent. angle dbc and angle adb form a pair of alternate interior angles. therefore, ad is congruent and parallel to bc. quadrilateral abcd is a parallelogram because its opposite sides are equal and parallel. which phrase best completes the students proof? aas postulate hl postulate sas postulate sss postulate

Explanation:

Step1: Identify given congruent parts

We know that $AB = DC$ (given), $\angle ABD=\angle CDB$ (alternate - interior angles as $AB\parallel DC$), and $DB$ is common to $\triangle ABD$ and $\triangle CDB$.

Step2: Recall congruence postulates

The Side - Angle - Side (SAS) postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, the two sides are $AB$ and $DB$ in $\triangle ABD$ and $DC$ and $DB$ in $\triangle CDB$, and the included angles $\angle ABD$ and $\angle CDB$ are congruent.

Answer:

C. SAS Postulate