Interior/Exterior Angles
We have learnt about the interior angles inside a triangle (do you remember they add to \(180^{\circ}\)?). In this lesson, we will expand that knowledge to different shapes (called polygons).
Interior Sum of Angles
The interior angles of any specific polygon always add up to a constant value, which depends only on the number of sides. In other words, regardless of how irregular the shape is, the sum of its internal angles will remain the same.
For example a pentagon has \(5\) sides, and its interior angles always add up to \(540°\). This is because its shape can be made from three triangles.
\(3 \times 180° = 540°\)
Let's take a look at the interior angles of different shapes:
| Shape | Sides (n) | Sum of Interior Angles | Equiangular Angle |
|---|---|---|---|
| Triangle | \(3\) | \(180°\) | \(60°\) |
| Quadrilateral | \(4\) | \(360°\) | \(90°\) |
| Pentagon | \(5\) | \(540°\) | \(108°\) |
| Hexagon | \(6\) | \(720°\) | \(120°\) |
| Heptagon | \(7\) | \(900°\) | \(128.57°\) |
| Octagon | \(8\) | \(1080°\) | \(135°\) |
Notice that when the side increase by one, the sum of interior angles increases by \(180^{\circ}\) (this is a linear relation!). This can be expressed algebraically as:
\(\text{Sum of Interior Angles} = (n−2) × 180°\)
If the shape is regular and each angle is the same (equiangular polygon), all angles can be determined algebraically as such:
\(\theta = (n−2) × \cfrac{180°}{n}\)
Exterior Sum of Angles
If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle. In the figure below, angle \(\boldsymbol{d}\) would be one of the exterior angle of the triangle.
Also, since we are extending a side of the polygon, that exterior angle must be supplementary to the polygon's interior angle. Together, the adjacent interior and exterior angles will add to \(180°\) (\(b+d=180^{\circ}\)).
The general rules for determining an exterior angle is:
\(\text{Exterior Angle} = 180° - \text{Interior Angle}\)
The Exterior Angles of a polygon always add up to \(360°\). This can be expressed algebraically as:
\(\text{Sum of Exterior Angles} = \text{Exterior Angle} × n = 360\)
This might seem strange, but think about the shape that is made if you start from one external angle and move on to the next until the end .... it makes a circle! And a circle is \(360°\).
Given that each exterior angles are the same for each vertex on the polygon (equiangular polygon), the exterior angle can be calculated using the following formula:
\(\phi=\cfrac{360}{n}\)
First, we can determine the sum of interior angles for a quadrilateral:
\(\text{Sum of Interior Angles} = (n−2) × 180°\)
\(\text{Sum of Interior Angles} = (4−2) × 180°\)
\(\text{Sum of Interior Angles} = 2 × 180°\)
\(\text{Sum of Interior Angles} = 360°\)
Next, we can determine the missing interior angle by finding the difference between \(360\) and the sum of the other angles:
\(360 = 98 + 120 + 108 + \theta\)
\(\theta = 34^{\circ}\)
Therefore, we can determine that the missing interior angle is \(\boldsymbol{34^\circ}\).
The sum of exterior angles is always 360°. If each exterior angle is 72°, then we can substitute this value into the corresponding formula and solve:
\(\text{Sum of Exterior Angles} = \text{Exterior Angle} × n = 360\)
\(\text{Sum of Exterior Angles} = 72n = 360\)
\(n = 5\)
Therefore, we can determine that the polygon has \(\boldsymbol{5}\) sides.
First, we can sketch a diagram of the triangle to help visualize the problem:
Next, we can determine the sum of interior angles for a triangle as such:
\(\text{Sum of Interior Angles} = (n−2) × 180°\)
\(\text{Sum of Interior Angles} = (3−2) × 180°\)
\(\text{Sum of Interior Angles} = 180°\)
We have one angle but two are missing. In order to solve for \(x\) we will need to solve for \(y\) first.
The question states the exterior angle to \(y\) is \(92^\circ\). Remember that the interior and adjacent exterior are supplementary and add to \(180^\circ\). As a result, we can determine \(y\) by finding the difference between \(180\) and \(\phi\):
\(\ 180 = y + \phi\)
\(\ 180 = y + 92\)
\(\ 180 = y + 92\)
\(y = 88\)
Finally, we can solve for the third angle, \(x\), by finding the difference between \(180\) and the sum of the other angles:
\(180 = 50 + 88 + x \)
\(x = 42^{\circ}\)
Therefore, we can determine that the value of the angle \(\boldsymbol{x = 42^\circ}\).