# How to Prove that a Quadrilateral Is a Rectangle

There are three ways to prove that a quadrilateral is a rectangle. Note that the second and third methods require that you first show (or be given) that the quadrilateral in question is a parallelogram:

If all angles in a quadrilateral are right angles, then it’s a rectangle (reverse of the rectangle definition). (Actually, you only need to show that three angles are right angles — if they are, the fourth one is automatically a right angle as well.)

If the diagonals of a parallelogram are congruent, then it’s a rectangle (neither the reverse of the definition nor the converse of a property).

If a parallelogram contains a right angle, then it’s a rectangle (neither the reverse of the definition nor the converse of a property).

*Tip:*

Before looking at any of these proof methods in action, here’s a useful little theorem that you need to do the upcoming proof.

**Congruent supplementary angles are right angles: **If two angles are both supplementary and congruent, then they’re right angles. This idea makes sense because 90° + 90° = 180°.

Okay, so here’s the proof:

**Statement 1****:**

*Reason for statement 1**:* Given.

**Statement 2****:**

*Reason for statement 2**:* If same-side exterior angles are supplementary, then lines are parallel.

**Statement 3****:**

*Reason for statement 3**:* If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.

**Statement 4****:**

*Reason for statement 4**:* If two angles are supplementary to the same angle, then they’re congruent.

**Statement 5****:**

*Reason for statement 5**:* Given.

**Statement 6****:**

*Reason for statement 6**:* If two angles are both supplementary and congruent, then they’re right angles.

**Statement 7****:**

*Reason for statement 7**:* If lines form a right angle, then they’re perpendicular.

**Statement 8****:**

*Reason for statement 8**:* If lines are perpendicular, then they form right angles.

**Statement 9****:**

*Reason for statement 9**:* If a parallelogram contains a right angle, then it’s a rectangle.

**Statement 10****:**

*Reason for statement 10**:* The diagonals of a rectangle are congruent.