# Which types of quadrilateral have exactly three right angles?

Jan 22, 2016

Quadrilaterals have $4$ sides and $4$ angles. The exterior angles of any convex polygon (ie no interior angle is less than $180$ degrees) add up to $360$ degrees ($4$ right angles). If an interior angle is a right angle then the corresponding exterior angle must also be a right angle (interior + exterior = a straight line = $2$ right angles).

Here $3$ internal angles are each right angles, so the corresponding $3$ external angles are also right angles, making a total of $3$ right angles. The remaining external angle must be $1$ right angle $\left(= 4 - 3\right)$, so the remaining $4 t h$ interior angle is also a right angle.

Therefore, if $3$ internal angles are right angles, the 4th angle must also be a right angle.

So no quadrilaterals have exactly $3$ right angles.

May 11, 2017

The types of quadrilaterals that have $3$ right angles are known as:
- Squares
- Rectangles
- Other shapes where all angles are ${90}^{o}$

#### Explanation:

The reason for this is:

All quadrilaterals interior angles must add up to exactly ${360}^{o}$.

So:

= $360 - \left(90 + 90 + 90\right)$
= $90$

And thus, the fourth angle must be ${90}^{o}$. The only quadrilaterals that fit the description where all angles are ${90}^{o}$ are squares and rectangles.

All the best!