# 5 Ways to Prove a Quadrilateral is a Parallelogram

This article will help us learn how to prove something is a parallelogram. More precisely, how to prove a quadrilateral is a parallelogram.

There are 5 basic ways to prove a quadrilateral is a parallelogram. They are as follows:

1. Proving opposite sides are congruent
2. Proving opposite sides are parallel
3. Proving the quadrilateral’s diagonals bisect each other
4. Proving opposite angles are congruent
5. Proving consecutive angles are supplementary (adding to 180Â°)

Let us now prove the above statements one by one.

### 1) Proving Opposite Sides are Congruent

Prove that opposite sides of a parallelogram are congruent

To prove: AB â‰… CD, AD â‰… BC

Proof:

Given: AB âˆ¥ CD, AD âˆ¥ BC

Draw in a diagonal AC

### 2) Proving Opposite Sides are Parallel

Prove that opposite sides of a parallelogram are parallel

To prove: AB âˆ¥ CD, AD âˆ¥ BC

Proof:

Given: AB â‰… CD, AD â‰… BC

Draw in a diagonal AC

### 3) Proving Diagonals Bisect Each Other

Prove that the diagonals of a parallelogram bisect each other.

This is an “if and only if” proof, so there are two things to prove.

1. To prove: AE â‰… EC, BE â‰… ED

Proof:

Given: ABCD is a parallelogram

And the converse

2. To prove: ABCD is a parallelogram

Proof:

Given: AE â‰… EC, BE â‰… ED

### 4) Proving Opposite Angles are Congruent

Prove that the opposite angles of a parallelogram are congruent.

To prove: âˆ ADC â‰… âˆ ABC, âˆ BAD â‰…  âˆ BCD

Proof:

Given: AB âˆ¥ CD, AD âˆ¥ BC

Draw in a diagonal AC

### 5) Proving Consecutive Angles are Supplementary

Prove that the consecutive angles of a parallelogram are supplementary (add up to 180Â°).

To prove: âˆ ABC + âˆ BCD = 180Â°

Proof:

Given: AB âˆ¥ CD, AD âˆ¥ BC

Extend BC till P