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2D shapes Triangles Parallelograms Types of angles Right angles Parallel linesHere you will learn about interior and exterior angles of polygons, including what interior and exterior angles of polygons are, and how to calculate angles in polygons using a variety of methods.

Students will first learn about interior and exterior angles of polygons as a part of geometry in high school.

**Interior and exterior angles of polygons** are two types of angles found in polygons. At any given vertex of a polygon, the interior and exterior angles are supplementary, meaning that they add together to equal 180^{\circ}.

A **polygon** is a two dimensional shape with at least three sides, where the sides of a polygon are all straight lines. βPolyβ comes from the Greek for βmanyβ and βgonβ means βanglesβ. You should be familiar with many types of polygons such as triangles, rectangles, pentagons, and heptagons.

A polygon where **all angles are equal size** and **all sides are equal length** is called a **regular polygon**.

A polygon where all angles are **not** equal size and/or all sides of the polygon are **not** equal length is called an **irregular polygon**.

Use this quiz to check your grade 4 studentsβ understanding of identifying and measuring angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

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DOWNLOAD FREE**Interior angles** are the angles within a polygon made by two adjacent sides.

You can calculate the sum of the interior angles of a polygon by subtracting 2 from the number of sides and then multiplying by 180^{\circ}.

The **sum of interior angles** of a regular polygon means finding the total of all the angles inside a polygon. The sum of all the angles in a triangle is equal to 180^{\circ}, but what about angles in a quadrilateral? A regular pentagon? Or even an irregular octagon?.

**Step by step guide:** Interior angles of a polygon

The three angles in any triangle add up to 180^{\circ}, or βThe sum of interior angles for a triangle is 180 degreesβ.

**Step by step guide: **Angles of a triangle

A quadrilateral is a four sided shape. A quadrilateral can be split into two triangles by drawing a line from one corner to an opposite one.

If the sum of interior angles of one triangle is 180^{\circ}, then the sum of the interior angles of two triangles is 180^{\circ}\times{2}=360^{\circ}

The sum of the interior angles of a quadrilateral is 360^{\circ}.

**Step by step guide:** Quadrilateral angles

Using the knowledge of triangles, you can find the sum of the interior angles of any polygon by splitting it into a number of triangles.

The interior angles of a polygon can also be found using the formula:

Sum of interior angles =(n-2)\times{180}

For example, a regular hexagon has 6 sides so n=6. You can substitute n=6 into the formula.

Sum of interior angles =(6-2)\times{180}=4\times{180}=720^{\circ}

The sum of the interior angles of a hexagon is 720^{\circ}.

**Step by step guide:**Β Angles of a hexagon

For **any regular polygon:**

\text{Each interior angle}=\cfrac{\text{Sum of interior angles}}{\text{Number of sides}}

Continuing with the regular hexagon above,

\text{Each interior angle}=\cfrac{\text{Sum of interior angles}}{\text{Number of sides}}=\cfrac{720}{6}=120^{\circ}

Each interior angle of a regular hexagon is 120^{\circ}.

**Exterior angles** are the angles between a polygon and the extended line from the next side.

The **sum of the exterior angles of a regular polygon** is always 360^{\circ} regardless of the number of sides the polygon has.

The exterior angles are angles between a polygon and the extended line from the vertex (center) of the polygon.

Sum of Exterior Angles of a Polygon =360^{\circ}

The interior angle and exterior angle at a vertex form a straight line, which adds to 180^{\circ}.

For a **regular polygon**, every exterior angle is the same and is calculated using the formula:

where E represents one exterior angle and n is the number of sides of the regular polygon.

**Step by step guide:** Sum of exterior angles of a polygon

How does this relate to high school math?

**High School: Geometry (HS.G.CO.C.10)**Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180^{\circ}; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

**High School: Geometry (HS.G.CO.C.11)**

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

**Identify the number of sides and whether any polygons are regular/irregular.****Find the sum of interior angles of the polygon(s).****Solve the problem using the information gathered.**

Find the sum of interior angles for this polygon.

**Identify the number of sides and whether any polygons are regular/irregular.**

This polygon has four sides.

This polygon is irregular as all the sides are not of equal length.

2**Find the sum of interior angles of the polygon(s).**

Sum of interior angles =(n-2)\times{180}

As the polygon has 4 sides, n=4, so you can substitute n=4 into the formula.

Sum of interior angles =(4-2)\times{180}=2\times{180}=360^{\circ}

3**Solve the problem using the information gathered.**

The question is asking for the sum of interior angles of the polygon.

Sum of interior angles =360^{\circ}

Find the size of each interior angle for a regular nonagon.

**Identify the number of sides and whether any polygons are regular/irregular.**

9 sides

Regular shape

**Find the sum of interior angles of the polygon(s) given.**

Sum of interior angles =(n-2)\times{180}

As a nonagon has 9 sides, n=9, so you can substitute n=9 into the formula.

Sum of interior angles of a nonagon =(9-2)\times{180}=7\times{180}=1260^{\circ}

**Solve the problem using the information gathered.**

The question is asking for βeach interior angleβ.

This means the size of one interior angle.

You know the sum of the interior angles for this polygon is 1260^{\circ}. Since it is a regular polygon, all the angles are of equal size.

Therefore, you can find the size of each interior angle by dividing the sum of interior angles by the number of angles in the polygon.

Each interior angle =\cfrac{1260}{9}=140^{\circ}

The diagram shows a convex polygon. Find the size of angle x.

**Identify the number of sides and whether any polygons are regular/irregular.**

6 sides

Irregular hexagon

**Find the sum of interior angles of the polygon(s) given.**

Sum of Interior Angles =(n-2)\times{180}

As n=6,

Sum of Interior Angles =(6-2)\times{180}=4\times{180}=720^{\circ}

**Solve the problem using the information gathered.**

You need to calculate the missing angle x. Note that you know the values of all the other angles.

\begin{aligned}150+90+105+130+145+x&=720 \\\\
620+x&=720 \\\\
x&=100 \end{aligned}

The size of angle x=100^{\circ}

**Identify the number of sides and whether any polygons are regular/irregular.****Identify what the question is asking and recall the sum of exterior angles.****Use the known information and any correct formula to solve.**

What is the size of one of the exterior angles of an equilateral triangle?

**Identify the number of sides and whether any polygons are regular/irregular.**

An equilateral triangle has 3 angles of equal size.

**Identify what the question is asking and recall the sum of exterior angles.**

The question is asking to find the size of one exterior angle of a triangle.

The sum of exterior angles for a triangle is 360^{\circ}.

**Use the known information and any correct formula to solve.**

As the shape is regular, each exterior angle is the same size. You can therefore divide the sum of exterior angles (360^{\circ}) by the number of slides (3) to get the size of one exterior angle.

360\div{3}=120^{\circ}

An irregular heptagon has one interior angle that is 115^{\circ}. What is the size of the adjacent exterior angle?

**Identify the number of sides and whether any polygons are regular/irregular.**

An irregular heptagon is a seven sided polygon.

**Identify what the question is asking and recall the sum of exterior angles.**

As adjacent means next to, you are being asked to find the size of the exterior angle where the interior angle on the line is known.

**Use the known information and any correct formula to solve.**

Angles on a straight line add to 180^{\circ} and the interior angle is 115^{\circ}.

180-115=65^{\circ}

The exterior angle will be 65^{\circ}.

The size of each interior angle of a regular polygon is 140^{\circ}. How many sides does the polygon have?

**Identify the number of sides and whether any polygons are regular/irregular.**

The polygon has an unknown number of sides, or n sides.

It is a regular polygon, therefore all exterior angles are equal.

**Identify what the question is asking and recall the sum of exterior angles.**

You need to find the number of sides.

The sum of the exterior angles of a polygon is 360^{\circ} and each exterior angle is equal because it is a regular polygon.

The sum of an interior and an exterior angle is 180^{\circ}.

**Use the known information and any correct formula to solve.**

If the interior angle is 140^{\circ} then the exterior angle will be 180-140=40^{\circ}.

The number of sides can therefore be calculated by 360\div{40}=9.

The polygon has 9 sides.

- Create anchor charts for the classroom that show the different polygons with the interior and exterior angles labeled. Using different colors can assist students in differentiating between the two types of angles.

- Instead of using worksheets and having students label the interior and exterior angles, have students use cut outs of the different polygons and have them investigate the measures of each.

**Confusing interior and exterior angles**

Reinforce with students that interior means βinsideβ and exterior means βoutsideβ, and they come together to form a side. Using labeled diagrams can help students with differentiating the two types of angles.

**Misidentifying if a polygon is regular or irregular**

Make it clear that regular polygons have equal interior and exterior angles, but irregular polygons do not. Teach students to recognize when a polygon is regular versus irregular and to adjust their calculations as needed.

**Forgetting that interior and exterior angles are supplementary**

Students need to be familiar that an exterior angle and interior angle add up to 180 degrees. Reinforce with practice problems where students are given either the interior or exterior angle and asked to find the other.

1. Find the size of each interior angle for a regular octagon.

1080^{\circ}

108^{\circ}

135^{\circ}

142^{\circ}

A regular octagon is an 8 -sided, regular shape, so you can substitute n=8 into the formula.

Sum of interior angles =(n-2)\times{180}=(8-2)\times{180}=6\times{180}=1080^{\circ}

The question is asking for βeach interior angleβ, which means the size of one interior angle.

You know the sum of the interior angles for this polygon is 1080^{\circ}. Since it is a regular polygon, all the angles are of equal size.

Each interior angle =\cfrac{1080}{8}=135^{\circ}

The size of each interior angle is 135^{\circ}.

2. Find the size of each interior angle for a regular decagon.

144^{\circ}

1440^{\circ}

180^{\circ}

160^{\circ}

A regular decagon is a 10 -sided, regular shape, so you can substitute n=10 into the formula.

\begin{aligned}\text{Sum of interior angles}&=(n-2)\times{180} \\\\ &=(10-2)\times{180} \\\\ &=8\times{180} \\\\ &=1440^{\circ} \end{aligned}

The question is asking for βeach interior angleβ, which means the size of one interior angle.

You know the sum of the interior angles for this polygon is 1440^{\circ}. Since it is a regular polygon, all the angles are of equal size.

\text{Each interior angle}=\cfrac{1440}{10}=144^{\circ}

The size of each interior angle is 144^{\circ}.

3. Find the size of angle y.

540^{\circ}

347^{\circ}

187^{\circ}

193^{\circ}

Find the sum of interior angles of the 5 -sided polygon given.

\begin{aligned}\text{Sum of interior angles}&=(n-2)\times{180} \\\\ &=(5-2)\times{180} \\\\ &=540^{\circ} \end{aligned}

\begin{aligned}42+45+125+135+y&=540 \\\\ 347+y&=540 \\\\ y&=193^{\circ} \end{aligned}

4. Find the size of one exterior angle of the regular polygon.

108^{\circ}

78^{\circ}

72^{\circ}

101^{\circ}

The polygon has 5 sides, and the sum of the exterior angles is 360^{\circ}.

Use the sum of the interior angles of a pentagon, 540^{\circ}.

\text{One interior angle}=540\div{5}=108^{\circ}Β

One exterior angle is equal to 180-108=72^{\circ} because the interior angle and exterior angle of a polygon lie on a straight line.

The measure of one exterior angle is 72^{\circ}.

5. An irregular hexagon has one interior angle that is 87^{\circ}. What is the size of the adjacent exterior angle?

87^{\circ}

93^{\circ}

107^{\circ}

273^{\circ}

As adjacent means next to, you are being asked to find the size of the exterior angle where the interior angle on the line is known.

Angles on a straight line add to 180^{\circ} and the interior angle is 87^{\circ}.

180-87=93^{\circ}

The exterior angle will be 93^{\circ}.

6. The size of each interior angle of a regular polygon is 120^{\circ}. How many sides does the polygon have?

8 sides

9 sides

5 sides

6 sides

The sum of the exterior angles of a polygon is 360^{\circ} and each exterior angle is equal because it is a regular polygon. The sum of an interior and an exterior angle is 180^{\circ}.

If the interior angle is 105^{\circ} then the exterior angle will be 180-120=60^{\circ}.

The number of sides can therefore be calculated by 360 \div 60=6.

The polygon has 6 sides.

The exterior angle theorem states that the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. In other words, the exterior angle is equal to the sum of the two interior angles that are not directly next to it.

No, because the interior and exterior angles are supplementary (add up to 180^{\circ} ), the exterior angle cannot be larger than the interior angle.

The smallest possible interior angle of a regular polygon is 60^{\circ}, which occurs in an equilateral triangle.

- Congruence and similarity
- Transformations
- Mathematical proof
- Trigonometry

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