1. ## parallelogram

If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent. What is the converse of this statement?
Is it true?

If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.

This is false.

Am I right?

2. Hello, Veronica1999!

If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent.
What is the converse of this statement? .Is it true?

If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
Yes, this is the converse.

This is false. . No, it is true.

We have quadrilateral $\displaystyle \,ABCD$ with $\displaystyle AB = CD,\;BC = AD.$

Draw diagonal $\displaystyle \,AC.$

Let $\displaystyle \angle 1 = \angle CAB,\;\angle 2 = \angle ACD.$

Code:

D * - - - - - * C
/      2  * /
/       *   /
/     *     /
/   *       /
/ *  1      /
A * - - - - - * B

In triangles $\displaystyle ABC$ and $\displaystyle CDA\!:\;AB = CD,\;BC = AD,\;AC = AC.$

Hence: .$\displaystyle \Delta ABC\,\cong\,\Delta CDA \quad\Rightarrow\quad \angle 1 \,=\,\angle 2$

Therefore: .$\displaystyle AB \parallel CD$

Since $\displaystyle AB = CD$ and $\displaystyle AB \parallel CD,\:ABCD$ is a parallelogram.

[Theorem: if two sides of a quadrilateral are equal and parallel,
. . . . . . . . the quadrilateral is a parallelogram.]