5.03 Polygon proofs

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)\times 180\degree, 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\degree. For n=3, the interior angle sum is 180\degree.

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\degree to the interior angle sum. For n=4, the interior angle sum is 360\degree.

If we add another point, we will get a pentagon. Again, doing so adds three new angles which sum to 180\degree. For n=5, the interior angle sum is 540\degree.

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.

In the square on the left, not all the vertices of the triangles are vertices of the square, so it has been divided incorrectly. In the square on the right, all the vertices of the triangles are vertices of the square, so it has been divided correctly.

Examples

Example 1

Consider the diagram below:

a

State the angle sum of a pentagon.

Worked Solution

Create a strategy

Use the formula for angle sum of a polygon: \left(n-2\right)\times 180\degree.

Apply the idea

Note that the polygon has 5 sides which means n=5.

\displaystyle \text{Angle sum}	\displaystyle =	\displaystyle (5-2) \times 180\degree	Substitute n=5
	\displaystyle =	\displaystyle 540\degree	Evaluate

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b

Solve for the value of x.

Worked Solution

Create a strategy

Equate the angle sum found from part (a) to the sum of all interior angles.

Apply the idea

To solve for x, we equate the angle sum of 540\degree to the sum of all interior angles.

\displaystyle x+126+122+92+96	\displaystyle =	\displaystyle 540	(Angle sum of a pentagon)
\displaystyle x+436	\displaystyle =	\displaystyle 540	Add all the constant terms
\displaystyle x+436-436	\displaystyle =	\displaystyle 540-436	Subtract 436 from both sides
\displaystyle x	\displaystyle =	\displaystyle 104\degree	Evaluate

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Example 2

Solve for x in the diagram below:

Worked Solution

Create a strategy

Use the formula for angle sum of a polygon: \left(n-2\right)\times 180\degree to find the interior angle. Then subtract this angle from 360\degree.

Apply the idea

Note that the polygon has 6 sides which means n=6.

\displaystyle \text{Angle sum}	\displaystyle =	\displaystyle (6-2) \times 180\degree	Substitute n=6
	\displaystyle =	\displaystyle 720\degree	Evaluate

We now equate the angle sum 720\degree to the sum of all interior angles. We can let the unknown interior angle be y.

\displaystyle 76+137+124+77+82+y	\displaystyle =	\displaystyle 720	(Angle sum of a hexagon)
\displaystyle 496+y	\displaystyle =	\displaystyle 720	Add the constant terms
\displaystyle y	\displaystyle =	\displaystyle 224	Subtract 496 from both sides

x and 224 are angles at a point, so they add up to 360\degree.

\displaystyle x+224	\displaystyle =	\displaystyle 360	(Angles in a revolution)
\displaystyle x	\displaystyle =	\displaystyle 136\degree	Subtract 224 from both sides

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

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\degree.

We can also see how this happens as we scale the shape down so that the vertices are closer together.

Again we can see the exterior angles make up a revolution.

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.

Examples

Example 3

Solve for x in the diagram below:

Show all working and reasoning.

Worked Solution

Create a strategy

Equate the sum of the exterior angles to 360\degree.

Apply the idea

\displaystyle x+132+127	\displaystyle =	\displaystyle 360	Add the exterior angles
\displaystyle x+259	\displaystyle =	\displaystyle 360	Add the constant terms
\displaystyle x+259-259	\displaystyle =	\displaystyle 360-259	Subtract 259 from both sides
\displaystyle x	\displaystyle =	\displaystyle 101\degree	Evaluate

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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.

Examples

Example 4

Solve for the value of \angle ABC in the diagram below, where ABFGH and BCDEF are regular pentagons:

Show all working and reasoning.

Worked Solution

Create a strategy

Find the interior angle sum of a pentagon using the formula \left(n-2\right)\times 180\degree, and divide it by the number of sides.

Apply the idea

\angle ABC, \, \angle ABF and \angle CBF are angles at a point, so they add up to 360 \degree.

ABFGH and BCDEF are congruent regular pentagons since they share a common side.

\angle ABF and \angle CBF are interior angles of congruent regular polygons, so we can divide the angle sum of a pentagon by 5 to find their size.

\displaystyle \text{Angle sum}	\displaystyle =	\displaystyle (5-2) \times 180\degree	Substitute n=5
	\displaystyle =	\displaystyle 540\degree	Evaluate

\displaystyle \angle ABF	\displaystyle =	\displaystyle \dfrac{540}{5}	(Angle of a regular pentagon)
	\displaystyle =	\displaystyle 108\degree
	\displaystyle =	\displaystyle \angle CBF	(Interior angles of congruent regular polygons)

\displaystyle \angle ABC + \angle ABF +\angle CBF	\displaystyle =	\displaystyle 360\degree	(Angles at a point)
\displaystyle \angle ABC + 108\degree +108\degree	\displaystyle =	\displaystyle 360\degree
\displaystyle \angle ABC	\displaystyle =	\displaystyle 360\degree-108\degree-108\degree
	\displaystyle =	\displaystyle 144\degree

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