Polygons are straightedged 2D shapes where the number of sides equals the number of vertices. Since this definition covers so many shapes, there aren't many properties that are common to all polygons.
However, when proving things involving polygons, there are a couple of features that we can rely on.
The interior angle sum of a polygon is always equal to
$\left(n2\right)\times180^\circ$(n−2)×180°
Where $n$n is the number of sides of the polygon.
The exterior angle sum of a convex polygon is always equal to $360^\circ$360°.
Let's have a look at why these are true.
The interior angle sum of a polygon is the sum of the angles inside the polygon. To show why the interior angle sum of a polygon is equal to $\left(n2\right)\times180^\circ$(n−2)×180°, we can think of polygons as collections of triangles.
Starting with a triangle, we know that it has three vertices and has an interior angle sum of $180^\circ$180°.
To make a quadrilateral, we can add another point. Doing this adds three new angles to the interior angle sum. Since these new angles are in a triangle, we have added $180^\circ$180° to the interior angle sum.
If we add another point, we will get a pentagon. Again, doing so adds three new angles which sum to $180^\circ$180°.



Since we can continue adding points in this way indefinitely, this rule holds for polygons with any number of vertices.
If we break polygons up into triangles to find the angle sum, each triangle's vertices need to be vertices of the polygon.


The exterior angles of a shape are the angles supplementary to the interior angles, determined by extending the sides of the polygon either clockwise or anticlockwise.



If we take all the exterior angles and place them together, we can see that they are angles around a point, so their sum must be $360^\circ$360°.
We can also see how this happens as we scale the shape down so that the vertices are closer together.



This property only applies to convex polygons. This is because exterior angles of nonconvex polygons can be inside the shape, causing angles to overlap when placed around a point.


Regular polygons are polygons which have all their sides equal in length. This also means that all their interior angles are equal.
Since all the interior angles of a regular polygon are equal, we can find the size of each interior angle by dividing the interior angle sum of the polygon by the number of angles.
What is the size of an interior angle in a regular hexagon?
Think: We can find the interior angle sum of a hexagon using the formula $\left(n2\right)\times180^\circ$(n−2)×180°. Since the hexagon is regular, the size of an interior angle will be this angle sum divided by $6$6.
Do: Using the formula, we find that the interior angle sum of the hexagon is:
$\left(62\right)\times180^\circ=720^\circ$(6−2)×180°=720°
Dividing this by the number of angles in the hexagon tells us that an interior angle in a regular hexagon has a size of:
$120^\circ$120°
Solve for $x$x in the diagram below:
Show all working and reasoning.
Solve for $x$x in the diagram below:
Show all working and reasoning.
Solve for $x$x in the diagram below:
Show all working and reasoning.
calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar
proves triangles are similar, and uses formal geometric reasoning to establish properties of triangles and quadrilaterals