Question

proving the parallelogram side theorem. given: abcd is a parallelogram. prove: $overline{ab}congoverline{cd}$ and $overline{bc}congoverline{da}$

Explanation:

Step1: Draw diagonal AC

In parallelogram ABCD, draw diagonal $\overline{AC}$.

Step2: Use properties of parallel lines

Since AB || CD and AD || BC (by definition of parallelogram), $\angle BAC\cong\angle DCA$ and $\angle ACB\cong\angle CAD$ (alternate - interior angles).

Step3: Prove triangle congruence

In $\triangle ABC$ and $\triangle CDA$, we have $\angle BAC\cong\angle DCA$, $\overline{AC}\cong\overline{CA}$ (reflexive property), and $\angle ACB\cong\angle CAD$. So, $\triangle ABC\cong\triangle CDA$ by the ASA (Angle - Side - Angle) congruence criterion.

Step4: Use congruent - triangle properties

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

Answer:

We have proved that in parallelogram ABCD, $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{DA}$.