Introduction
Polygons can be used to create amazing patterns. Investigating the angles inside and outside polygons can help us to understand their special qualities. In this unit we will look into the distinctive angular properties of different types of quadrilaterals and polygons.
Polygons can be used to create amazing patterns. Investigating the angles inside and outside polygons can help us to understand their special qualities. In this unit we will look into the distinctive angular properties of different types of quadrilaterals and polygons.
Interior angles in quadrilaterals
If we take any foursided
polygon
A polygon is a closed, planar figure bounded by straight line segments.polygon, we can split it into two triangles. To do this, we just join a set of opposite vertices with a
straight line
A straight line is a set of points related by an equation of the form y = ax + c. It has length and position, but no breadth and is therefore onedimensional.straight line. Fig.1 below shows how this is done in the case of a
parallelogram
A parallelogram is a quadrilateral with two sets of parallel sides.parallelogram, a
trapezium
A trapezium is a quadrilateral with one set of parallel sides and one set of nonparallel sides.trapezium, and a
rectangle
A rectangle is a quadrilateral with four interior angles of 90.rectangle.We know that the angles in a
triangle
A triangle is a threesided polygon.triangle add up to 180. We can also see that, together, the angles in each pair of triangles form the angles of each
quadrilateral
A quadrilateral is a polygon with four sides.quadrilateral. So the angles in each quadrilateral must sum to 360.Exterior angles in quadrilaterals
Look at the quadrilateral in Fig.4 below. All of its sides have been extended and its interior (blue) and exterior (red) angles marked.Notice that each exterior/interior angle pair lie on a straight line. Each pair must therefore sum to 180.
In any quadrilateral, there will be four such pairs. So we can see that:

However, we already know that the sum of the interior angles is 360, so:

So the exterior angles of a quadrilateral sum to 360.
The parallelogram
Many engineering designs rely on the parallelogram and its properties. For instance, construction vehicles such as that
below use the parallelogram shape to allow movement and stability.
Using information about
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.parallel lines,
perpendicular
Two lines or planes are perpendicular if they are at right angles to one another.perpendicular lines, and angles, it is possible to work out all the interior and exterior angles of parallelogram from just one given angle.What do you notice about the angles that you have entered? You should find that in the whole table, each angle has one of only two values.
If we look specifically at the interior angles s, t, u, and v, then we see that the opposite interior angles in the parallelogram are equal. Also, any two adjacent interior angles sum to 180.
You can see these relationships between the interior angles in a parallelogram in Fig.8. Drag a side or corner to see how the angles are related.
Click on the figure below to interact with the model.
We can show exactly why these angle patterns exist by using the model in Fig.9 below. Click on and drag the blue angle in Fig.9. If you place the intersecting lines that form the blue angle over each corner of the parallelogram in turn you will see that each intersection is the same. We can say that the intersections are congruent.
Click on the figure below to interact with the model.
Interior angles in other polygons
The same methods that we use to look at quadrilaterals can be applied to other polygons, such as irregular pentagons and
hexagons. In Fig.11, an irregular
octagon
An octagon is an eightsided polygon.octagon has been divided up into triangles to help investigate its interior angles. We can divide the irregular
hexagon
A hexagon is a sixsided polygon.hexagon and the irregular
pentagon
A pentagon is a fivesided polygon.pentagon into triangles in the same way by joining vertices with lines inside the shapes only.Click on the figure below to interact with the model.
Try dragging the corners of the three shapes to see how the triangles move.
So the sum of the pentagon's interior angles is:

Likewise, the sum of the hexagon's interior angles is:

In the examples we have looked at, the number of triangles has been two fewer than the number of sides in the shape. In fact, this is the case for any polygon.

This expression can be rearranged as follows:
We can show this visually by dividing polygons into triangles in a different way. If we place a
point
 A point has no properties except position. It is an object with zero dimensions.
 Points in the xy plane can be specified using x and y coordinates.
vertex
vertex to this point with a straight line (3).
We have created 6 triangles with angles as shown (4). So the sum of the triangles' angles is .
However, six of the triangles' angles are formed around the central point (5). These sum to 360, but do not contribute to the interior angles of the hexagon. So here:

We can apply the same reasoning to any type of polygon. In every case, each side of the polygon forms the base of one triangle. So if n is the number of sides the polygon has, n triangles are formed.
So, generally:

Exterior angles in other polygons
The exterior angles of any
convex
A polygon is convex if all of its interior angles are less than 180.convex polygon sum to 360. Fig.14 below illustrates one way of understanding this.We can explain this fact mathematically, using expressions that we have studied earlier. As we saw when we looked at quadrilaterals, each of the interior and
exterior angle
When the side of a convex polygon is produced (lengthened), the exterior angle is the angle between this line and an adjacent
side.exterior angle pairs adds up to 180°, as together they make up a straight line. So the total of the interior and exterior angles in a convex polygon is equal to the number of straight lines forming the polygon multiplied by 180°.So the interior and exterior angles of the above
heptagon
A heptagon is a sevensided polygon.heptagon sum to:

So, generally, the interior and exterior angles of an nsided polygon sum to:

Since we know a general expression for the sum of interior angles of an nsided polygon, we can work out an expression for the exterior angle sum:
So we have found that the exterior angles of a convex polygon sum to 360, regardless of the number of sides it has.
Summary
Splitting polygons up into triangles is a useful way of finding out about their interior angles.
The interior angles of a quadrilateral sum to 360.
The exterior angles of convex polygon sum to 360.
The interior angles of an nsided polygon sum to .
Splitting polygons up into triangles is a useful way of finding out about their interior angles.
The interior angles of a quadrilateral sum to 360.
The exterior angles of convex polygon sum to 360.
The interior angles of an nsided polygon sum to .
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