Interior angles in a quadrilateral
This fact is a more specific example of the equation for calculating the sum of the interior angles of a polygon:
\(\text {Sum of interior angles = (n – 2)} ~\times~180^\circ\)
where \(\text{n}\) is the number of sides.
Another way to calculate the sum of the interior angles of a polygon is to see how many triangles the shape is composed of. Quadrilaterals are composed of two triangles. Seeing as we know the sum of the interior angles of a triangle is 180°, it follows that the sum of the interior angles of a quadrilateral is 360°.
If we are given a shape with only one missing angle, we can use the other angles to calculate what value the missing angle should take.
Example
Find angle \({d}\).
Solution
\({d}~=~{360}~-~{100}~-~{70}~-~{30}~=~160^\circ\)
Often we can also use the other properties of a shape to work out more missing angles.
Use the fact that parallelograms have two pairs of equal angles to calculate the missing angles in this question.
Question
Find the unknown angles.
The quadrilateral is a parallelogram. Opposite angles are equal.
\({g}~=~50^\circ\)
Angles in a quadrilateral add up to 360° and opposite angles are equal.
\({f}~=~\frac {360~-~50~-~50} {2} = 130^\circ\)
In the equation above, we are dividing by 2 because 360 - 50 - 50 finds us the value of both \({f}\) and its equal partner (opposite at \({s}\))
Exterior angles
The exterior angles of a polygon always add up to 360°. Furthermore the interior and exterior angles at a point always add up to 180°.
An exterior angle is the angle made with the side of the shape, if you were to extend the side of the shape in one direction at each vertex.
Example
Find the interior angles of the shape below.
As we know, the exterior angle plus the interior angle adds up to 180°, so the missing angles moving clockwise from the top are:
- 180 – 71.4 = 108.6°
- 180 – 56.4 = 123.6°
- 180 – 111.5 = 68.5°
- 180 – 120.7 = 59.3°