Question

Prove that opposite sides of a parallelogram are congruent. Be sure to create and name the appropriate geometric figures.

Answer 1

Theorem : Opposite sides of a parallelogram are congruent or equal.

Let us suppose a parallelogram ABCD.  

Given:$AB\parallel CD$ and $BC\parallel AD$ (According to the definition of parallelogram)

We have to prove that: AB is congruent to CD and BC is congruent to AD.

Prove: let us take two triangles, $\bigtriangleup ACD$ and$\bigtriangleup ABC$

In these two triangles, $\angle1=\angle2$ { By the definition of alternative interior angles}

Similarly,                        $\angle4=\angle3$

And,                                AC=AC (common segment)

By  ASA, $\bigtriangleup ACD \cong \bigtriangleup ABC$

thus By the property of congruent triangle, we can say that corresponding  sides of $\bigtriangleup ACD and \bigtriangleup ABC$ are also congruent.

Thus, AB is congruent to CD and BC is congruent to AD.