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Wize High School Geometry Textbook (Common Core) > Polygons

Interior Angles of Polygons

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Definitions

A polygon is a shape that has multiple sides.
  • Triangles have 3 sides
  • Quadrilaterals have 4 sides
  • n-sided polygons have n sides

An interior angle of a polygon is an angle that is inside of the polygon.
A regular n-gon is a polygon with n equal sides and n equal interior angles (the exterior angles will also all be equal).
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Angles in a Triangle

The 3 interior angles of a triangle must add up to 180°180\degree!

Example
Determine the value of xx in the following triangle.


We know that all 3 interior angles of a triangle must add up to 180°180\degree:

x+40°+80°=180°x+120°=180°x=180°−120°x=60°\begin{array}{rcl} x+40\degree+80\degree&=&180\degree\\[1em] x+120\degree&=&180\degree\\[1em] x&=&180\degree-120\degree\\[1em] x&=&60\degree \end{array}
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Example: Angles in a Triangle

Determine the value of yy in the following triangle.

All 3 interior angles of a triangle must add up to 180°180\degree.

50°+2y+3y=180°50°+5y=180°5y=180°−50°5y=130°y=130°5y=26°\begin{array}{rcl} 50\degree+2y+3y&=&180\degree\\[1em] 50\degree+5y&=&180\degree\\[1em] 5y&=&180\degree-50\degree\\[1em] 5y&=&130\degree\\[1em] y&=&\dfrac{130\degree}{5}\\[1em] y&=&26\degree \end{array}
So, y=26°\boxed{y=26\degree}.

Practice: Angles in a Triangle

Find all missing angles in the following triangles.

a)




b)


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Interior Angles in a Polygon

Interior Angles in a Quadrilateral

Notice that any quadrilateral can be split up into two triangles.

Write it Down
What can we say about the sum of the interior angles in any quadrilateral?

The interior angles in any quadrilateral must add up to 2×180°=360°2\times180\degree=360\degree


Example 1
What is the sum of the interior angles of a square? A rectangle? A parallelogram?

They are all 360°360\degree! Since squares and rectangles have 4 right angles as interior angles, we know that the interior angles add up to 4×90°4\times90\degree, which is also 360°360\degree.

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Interior Angles in a Polygon

Example 2
If we try to split up a 5-sided polygon into triangles,

a) what is the least number of triangles you will need?

You will get 3 triangles, for example:

b) what can we say about the sum of the interior angles in a 5-sided polygon?

The sum of the interior angles must be 3×180°=540°3\times180\degree=540\degree


Example 3
If we try to split up a 8-sided polygon into triangles,

a) what is the least number of triangles you will need?

You will get 6 triangles, for example:

b) what can we say about the sum of the interior angles in a 8-sided polygon?

The sum of the interior angles must be 6×180°=1080°6\times180\degree=1080\degree


Write it Down
The sum of the interior angles in any n-sided polygon is
(n−2)×180°\left(n-2\right)\times180\degree

Practice: Interior Angles in a Polygon

a) How many sides does the above polygon have?

b) What is the sum of interior angles for the polygon above?

c) Determine the missing angle.

Practice: Interior Angles in a Polygon

What is the measure of one of the interior angles in a regular 20-sided polygon?

Practice: Interior Angles in a Polygon

If each of the interior angles of a regular polygon is 150°150\degree, how many sides does this regular polygon have?