Question

proving the parallelogram side theorem
given: abcd is a parallelogram
prove: ab ≅ cd and bc ≅ da
unique line postulate
statements reasons
given def of parallelogram
def of all interior angles
always start with the given information.

Explanation:

Step1: Draw diagonal AC

In parallelogram ABCD, draw diagonal AC.

Step2: Prove triangle congruence

In $\triangle ABC$ and $\triangle CDA$, $\angle BAC\cong\angle DCA$ (alternate - interior angles as $AB\parallel CD$), $AC = CA$ (common side), $\angle BCA\cong\angle DAC$ (alternate - interior angles as $BC\parallel DA$). So, $\triangle ABC\cong\triangle CDA$ by ASA (Angle - Side - Angle) congruence criterion.

Step3: Use congruent - triangle properties

Since $\triangle ABC\cong\triangle CDA$, then $AB\cong CD$ and $BC\cong DA$ (corresponding parts of congruent triangles are congruent).

Answer:

The proof shows that in parallelogram ABCD, $AB\cong CD$ and $BC\cong DA$ by using the properties of parallel lines to get congruent angles, a common side, and the ASA congruence criterion for triangles, and then the fact that corresponding parts of congruent triangles are congruent.