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9. Polygons
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Geometry

9.01 Angles of polygons

Lesson
Practice

Interior angles in polygons

A polygon is a closed plane figure formed by at least three straight sides. Polgons can be classified in many ways.

Convex polygon

A polygon with each interior angle measuring less than 180 \degree. All diagonals of a convex polygon lie inside the polygon.

A pentagon with all of its diagonals lies inside the pentagon.
Concave polygon

A polygon with at least one interior angle that has a measure greater than 180 \degree

A concave polygon showing an angle greater than 180 degrees.
Diagonal of a polygon

A line segment that connects the nonconsecutive vertices of a polygon

A quadrilateral showing two line segments each connecting a pair of non-adjacent vertices. The two line segments intersect each other inside the quadrilateral.

A regular polygon is a convex polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent).

A triangle with an auxiliary line is drawn on one of the vertices and parallel to the opposite side of the triangle. Three angles were formed from the auxiliary line and the vertex. Each non-parallel side of the triangle acts like a transversal of the two parallel lines. 2 Pairs of alternate interior angles are marked as equal.

Recall that the Triangle angle sum theorem states the sum of the measures of the interior angles of a triangle is 180\degree.

Exploration

Consider the quadrilateral shown:

A quadrilateral.
  1. What ways could you break the quadrilateral into the least number of triangles? How many triangles does this create?
  2. Determine the sum of the interior angles of the quadrilateral.
  3. Draw a hexagon and determine the least number of triangles you could break the polygon into, and determine the sum of its interior angles.
  4. What can you say about the relationship between the number of sides on any polygon, the number of triangles it can be divided into, and the sum of its interior angles?

The sum of the interior angle measures of a polygon depends on the number of sides of the polygon. A polygon with n sides (or an n-gon) can always be divided into (n-2) non-overlapping triangles. This fact and the triangle angle sum theorem helps us calculate interior angle sums and individual angle measures of regular polygons.

Polygon interior angle sum theorem

The sum of the measures of the interior angles of a convex n-gon is equal to \left(n-2\right)180 \degree

Corollary to the polygon interior angle sum theorem

The measure of each interior angle of a regular n-gon is \dfrac{\left(n-2\right)180 \degree}{n}

Examples

Example 1

Consider the polygon angle sum theorem.

a

Prove the interior polygon angle sum theorem works for a pentagon.

Worked Solution
Create a strategy

First, draw a pentagon with diagonals drawn from a vertex.

Pentagon A B C D E with diagonals A D and A C drawn.
Apply the idea

Given a pentagon with the diagonals drawn from a vertex, we can identify the number of triangles in the pentagon. The pentagon is broken into 3 triangles, and the sum of the interior angles of each triangle is 180 \degree. Using this information and the angle sum addition postulate, we know the sum of the interior angles of the pentagon is (5-2) \cdot 180 \degree = 3 \cdot 180 \degree = 540 \degree.

Reflect and check

We can use another polygon to prove the polygon angle sum theorem, such as an octagon.

An octagon with diagonals drawn from one vertex to the other vertices. The octagon is split into six non-overlapping triangles.

Draw an octagon with the diagonals drawn from one vertex. The total number of triangles drawn is 6. We know that the sum of the interior angles of each triangle is 180 \degree, so the sum of the angles must be 1080 \degree. This also follows from (8-2) \cdot 180 \degree = 6 \cdot 180 \degree = 1080 \degree.

b

Explain why the polygon angle sum theorem will work for any convex polygon.

Worked Solution
Create a strategy

We will use a convex polygon and draw triangles to conceptualize the theorem.

Apply the idea
A polygon with n number of sides that shows how many non-overlapping triangles the polygon can be split into. A polygon with n sides has n minus two non-overlapping triangles.

Consider a convex polygon with n sides and n vertices. To divide any n-gon into non-overlapping triangles, we will start with a random vertex, A_1, and draw a diagonal to every other non-adjacent vertex. A1 is adjacent to A_2 and A_n, so those diagonals are already part of the polygon. The diagonal between A_1 and A_3 creates the first triangle, the diagonal from A_1 to A_4 creates the second triangle, and this pattern continues until we connect A_1 to A_{n-1}, creating the last triangle. Thus, we will have created \left(n-2 \right) non-overlapping triangles. By the Triangle Angles Sum Theorem, we know that the sum of the interior angles of a triangle is 180 \degree, so the sum of the interior angles of (n-2) triangles is \left(n-2 \right) \cdot 180 \degree.

Example 2

For a regular 24-gon:

A regular 24 gon.
a

Find the sum of the interior angles.

Worked Solution
Create a strategy

We use the polygon angle sum theorem to find the sum of the interior angles. The polygon angle sum theorem states that the sum of the interior angles of a convex polygon is equal to \left (n-2\right)180 \degree for n sides.

A 24-gon is a convex polygon and has 24 sides. So we want to substitute 24 for n in the expression and evaluate.

Apply the idea

\left(24 - 2 \right)180 \degree= 3960 \degree

The sum of interior angles is 3960 \degree

b

Find the measure of a single interior angle.

Worked Solution
Create a strategy

Since we want to find the measure of a specific interior angle of a regular polygon, we want to use the corollary to the polygon angle sum theorem. This theorem states that the interior angle is equal to the sum of the interior angles divided by the number of sides, \dfrac{\left(n-2\right)180 \degree}{n}

Now we need to substitute 24 for n and evaluate.

Apply the idea

\dfrac{\left(24-2\right)180 \degree}{24}=165 \degree

The measure of an interior angle is 165 \degree

Reflect and check

Note that we can only find the measure of the interior angle because this is a regular polygon and all angles are the same. If it was a non-regular 24-gon, the interior angle sum would still be the same, but each interior angle could be a different measure.

Example 3

Find the value of y.

A quadrilateral with a right angle, a 57 degree angle, a 55 degree angle, and an unknown angle labeled y.
Worked Solution
Create a strategy

We know the measures of 3 angles in the given quadrilateral, so we can use the polygon angle sum theorem to determine the sum of the interior angles and write and solve an equation to find y.

Apply the idea
\displaystyle (n-2)180\displaystyle =\displaystyle (4-2)180Polygon angle sum theorem for a quadrilateral
\displaystyle =\displaystyle 360Evaluate the parentheses and multiplication

Since the sum of the interior angles of a quadrilateral are 360 \degree, we have

\displaystyle 57 + 90 + 55 + y\displaystyle =\displaystyle 360Sum of the interior angles of the quadrilateral
\displaystyle 202 + y\displaystyle =\displaystyle 360Combine like terms
\displaystyle y\displaystyle =\displaystyle 158Subtract 202 from both sides

y=158 \degree.

Example 4

Determine the number of sides of a regular polygon when each interior angle has a measure of 150\degree.

Worked Solution
Create a strategy

To find the number of sides of the regular polygon, we can use the corollary to the polygon angle sum theorem: I = \dfrac{\left(n - 2\right)180 \degree}{n} where I is the measure of each interior angle and n is the number of sides. We can substitute the given interior angle measure and solve for n.

Apply the idea

Given that each interior angle has a measure of 150\degree, we can substitute this value into the equation and solve for n:

\displaystyle 150\degree\displaystyle =\displaystyle \dfrac{(n - 2)180\degree}{n}Substitute I = 150\degree
\displaystyle 150\degree n\displaystyle =\displaystyle (n - 2)180\degreeMultiply both sides by n
\displaystyle 150\degree n\displaystyle =\displaystyle 180\degree n - 360\degreeDistribute 180\degree
\displaystyle -30\degree n\displaystyle =\displaystyle -360 \degreeSubtract 180\degree n from both sides
\displaystyle n\displaystyle =\displaystyle 12Divide by -30 \degree on both sides

This means that the regular polygon has 12 sides.

Reflect and check

We can check our answer by substituting n=12 back into the formula for the measure of each interior angle of a regular polygon:

\displaystyle I\displaystyle =\displaystyle \dfrac{(12 - 2)180\degree}{12}Substitute n=12
\displaystyle I\displaystyle =\displaystyle \dfrac{\left(10\right) 180\degree}{12}Evaluate the subtraction
\displaystyle I\displaystyle =\displaystyle 150\degreeEvaluate the multiplication and division

Since the calculated interior angle measure matches the given value of 150\degree, our answer of 12 sides for the regular polygon is correct.

Idea summary

We can use the polygon angle sum theorem and its corollary to find unknown angles of convex and regular n-gons:

  • The sum of the interior angles of a convex n-gon is equal to \left(n-2\right)180 \degree
  • The measure of each interior angle of a regular n-gon is \dfrac{\left(n-2\right)180 \degree}{n}

Exterior angles in polygons

Exploration

Drag the points to shape the convex polygon. Then, explore the applet.

Loading interactive...
  1. How would you describe the angles measured in the applet?
  2. What do you notice about the angle measurements?
A triangle showing the three interior and three exterior angles. Ask your teacher for more information

We can take any side of a polygon and extend it to create an exterior angle.

Two images are depicted. The first image illustrates a linear pair formed by an interior angle and its adjacent exterior angle of a triangle. In the second image, the three exterior angles of the triangle are combined, resulting in a sum of 360 degrees, representing one full revolution.

The exterior angle and the corresponding interior angle form a linear pair (add to 180\degree).

The sum of all of the exterior angles of a polygon is 360 \degree.

Polygon exterior angle sum theorem

The sum of the exterior angles of a convex polygon is 360 \degree

Corollary to the polygon exterior angle sum theorem

The measure of each exterior angle of a regular n-gon is \dfrac{1}{n} \cdot 360 \degree

Examples

Example 5

Prove the polygon exterior angle sum theorem.

Worked Solution
Create a strategy

The theorem states that the sum of the exterior angles of any polygon is 360 \degree. We already know the sum of the interior angles is 180\left(n-2\right) for an n-sided polygon.

Apply the idea

Let N be the sum of the exterior angles of an n-sided polygon. Let's focus on one exterior angle first for an n-sided polygon.

A polygon showing an exterior angle labeled x sub 1 degrees.

For this exterior angle with measure x_1 \degree, we can see that it forms a linear pair with the related interior angle. This means the interior angle would measure \left(180 - x_1\right) \degree.

This will be true for each exterior-interior angle pair, so for any exterior angle measuring x_n, the interior angle will be \left (180 - x_n \right).

So we have:

  • Sum of the exterior angles: N=x_1+x_2+\ldots +x_n
  • Sum of the interior angles that they form linear pairs with exterior angles:
    \displaystyle S\displaystyle =\displaystyle \left(180 - x_1\right)+\left(180 - x_2\right)+\ldots +\left(180 - x_n\right)
    \displaystyle S\displaystyle =\displaystyle 180n - \left(x_1+x_2+\ldots +x_n\right)Commutative, associative, and distributive properties of multiplication
    \displaystyle S\displaystyle =\displaystyle 180n - NSubstitution
  • Sum of interior angles using polygon angle sum theorem: S=180\left(n-2\right)

Putting together the two ways to express the interior angle sum we get:

\displaystyle S\displaystyle =\displaystyle 180(n-2)
\displaystyle 180\left(n-2\right)\displaystyle =\displaystyle 180n - NTransitive property of equality
\displaystyle 180n -360\displaystyle =\displaystyle 180n - NDistributive property
\displaystyle -360\displaystyle =\displaystyle -NSubtract 180n from both sides
\displaystyle 360\displaystyle =\displaystyle NMultiply both sides by -1

So the sum of the exterior angles of a polygon is 360 \degree.

Example 6

Determine the value of y:

A pentagon with some of its interior and exterior angles shown. The exterior angles are: 60 degrees, 42.5 degrees, and 3 y plus 18 degrees. The interior angles are: 112 degrees and 115 degrees. Speak to your teacher for more details.
Worked Solution
Create a strategy

We want to solve for y which is part of an expression of an exterior angle. To do so, we can use the polygon exterior angle theorem to write an equation relating the exterior angles of the polygon after calculating the measures of the exterior angles.

Apply the idea

The exterior angle that is supplementary to 115 \degree must be 65 \degree, and the exterior angle that is supplementary to 112 \degree must be 68 \degree.

\displaystyle 60 + 65 + 42.5 + (3y+18) + 68\displaystyle =\displaystyle 360Exterior angle formula
\displaystyle 3y+ 253.5\displaystyle =\displaystyle 360Combine like terms
\displaystyle 3y\displaystyle =\displaystyle 106.5Subtract 253.5 from both sides
\displaystyle y\displaystyle =\displaystyle 35.5Divide both sides by 3

Example 7

Determine the number of sides of a regular polygon when each exterior angle has a measure of 20\degree.

Worked Solution
Create a strategy

To find the number of sides of the regular polygon, we can use the corollary to the polygon exterior angle sum theorem: E = \dfrac{1}{n}\cdot 360\degree where E is the measure of each exterior angle and n is the number of sides. We can substitute the given exterior angle measure and solve for n.

Apply the idea

Given that each exterior angle has a measure of 20\degree, we can substitute this value into the equation and solve for n:

\displaystyle 20\degree\displaystyle =\displaystyle \dfrac{1}{n}\cdot360\degreeSubstitute E = 20\degree
\displaystyle 20\degree n\displaystyle =\displaystyle 360\degreeMultiply both sides by n
\displaystyle n\displaystyle =\displaystyle 18Divide both sides by 20\degree

This means that the regular polygon has 18 sides.

Reflect and check

We can check our answer by substituting n=18 back into the equation for the measure of each exterior angle of a regular polygon:

\displaystyle E\displaystyle =\displaystyle \dfrac{1}{18}\cdot 360\degreeSubstitute n=18
\displaystyle E\displaystyle =\displaystyle 20\degreeEvaluate

Since the calculated exterior angle measure matches the given value of 20\degree, our answer of 18 sides for the regular polygon is correct.

Idea summary

We can use the polygon exterior angle sum theorem and its corollary to find unknown angles of convex and regular n-gons:

  • The sum of the exterior angles of any polygon is 360 \degree
  • The measure of each exterior angle of a regular n-gon is \dfrac{1}{n} \cdot 360 \degree

Outcomes

G.PC.2

The student will verify relationships and solve problems involving the number of sides and angles of convex polygons.

G.PC.2a

Solve problems involving the number of sides of a regular polygon given the measures of the interior and exterior angles of the polygon.

G.PC.2b

Justify the relationship between the sum of the measures of the interior and exterior angles of a convex polygon and solve problems involving the sum of the measures of the angles.

G.PC.2c

Justify the relationship between the measure of each interior and exterior angle of a regular polygon and solve problems involving the measures of the angles.

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