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Stage 5.1-3

5.03 Polygon proofs

Lesson

Polygons are straight-edged 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.

Interior angle sum

The interior angle sum of a polygon is always equal to

$\left(n-2\right)\times180^\circ$(n2)×180°

Where $n$n is the number of sides of the polygon.

Exterior angle sum

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.

 

Interior angle sum of a polygon

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 $\left(n-2\right)\times180^\circ$(n2)×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°.

For $n=3$n=3, the interior angle sum is $180^\circ$180°.

For $n=4$n=4, the interior angle sum is $360^\circ$360°.

For $n=5$n=5, the interior angle sum is $540^\circ$540°.

Since we can continue adding points in this way indefinitely, this rule holds for polygons with any number of vertices.

Caution

If we break polygons up into triangles to find the angle sum, each triangle's vertices need to be vertices of the polygon.

Not all the vertices of the triangles are vertices of the square.

All the vertices of the triangles are vertices of the square.

 

Exterior angle sum of a 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 anti-clockwise.

Supplementary angles made by extending the sides anti-clockwise are exterior angles.

Supplementary angles made by extending the sides anti-clockwise are exterior angles.

All exterior angles must be made by extending sides in the same direction.

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.

Scaling down the shape.

Scaling down the shape some more.

Scaling the shape down to a point.

Caution

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

All exterior angles are outside the polygon.

Not all exterior angles are outside the polygon.

 

Regular polygons

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.

Worked example

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 $\left(n-2\right)\times180^\circ$(n2)×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:

$\left(6-2\right)\times180^\circ=720^\circ$(62)×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°

 

Practice questions

Question 1

Solve for $x$x in the diagram below:

  1. Show all working and reasoning.

Question 2

Solve for $x$x in the diagram below:

  1. Show all working and reasoning.

Question 3

Solve for $x$x in the diagram below:

  1. Show all working and reasoning.

Outcomes

MA5.2-14MG

calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar

MA5.3-16MG

proves triangles are similar, and uses formal geometric reasoning to establish properties of triangles and quadrilaterals

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