Solve problems using angle and symmetry properties of polygons
A polygon is a shape with straight sides. The strange name comes from Greek, where Poly means many, and gonia means angle (polygon: manyangle).
There are several rules about both the interior and exterior angles in polygons, as well as their symmetrical properties (lines of symmetry and rotational symmetry). It’s possible to use these rules to find unknown information (e.g. missing angles), and in your exam you will be expected to do this.
Angles in a Polygon
Regular polygons have sides of equal length and all the interior angles are equal.
If n is the number of sides a regular polygon has (n2) x 180^{o} is the sum of the interior angles in any polygon.
We can use the fact that all angles in a regular polygon are the same to calculate one interior angle. Since they are all the same we just divide the sum by n (the number of sides the polygon has).
The exterior angle is 180^{o} minus the interior angle.
How did we get the formula for the sum of interior angles?
If you draw any size polygon shape we can divide it into triangles (just draw lines connecting the corners to make triangles)
We know that the interior angles on a triangle add up to 180^{o}. So to calculate the sum of interior angles we just multiply the number of triangles we have by 180^{o}.
If you do this for polygons of different sizes, you will notice that the number of triangles will always be (n2).
For example
A regular pentagon has 5 sides. So the sum of interior angles is 3 x 180^{o} = 540^{o}. Every interior angle is 540^{o}/5 = 108. Every exterior angle is 180^{o}108^{o}=72^{o}.
If an irregular pentagon has 4 labelled interior angles that total 512^{o}, the last interior angle is 540512=28^{o}.
Symmetry
A line of symmetry means either side of the line are identical, this includes the length of all sides and the size of all angles.
It is possible to have more than 1 line of symmetry. It is also possible to have no lines of symmetry.
To help check if it's a line of symmetry or not, use a mirror to test it, or sketch it out and fold the paper.
Rotational Symmetry
How many times can you rotate a shape such that after rotation it looks the same as when you started? We call this number the order of rotation.
If a shape cannot be rotated to be the same, it has no rotational symmetry but the order is 1 (you can turn it 360^{o}!)
For example
A square has 4 lines of symmetry, and its order of rotation is 4.
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