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Outside the Box Geometry

# Polygon Angles

Sometimes, it’s possible to find the sum of several angles, even when you wouldn’t be able to calculate each angle in the sum individually.

What is the sum of the two red angles in this figure?

The three internal angles of a triangle always adding to $$180^\circ$$ is one of the most fundamental examples of being able to calculate the sum of a set of angles, even without knowing the individual measures. But what about other polygons or more complex figures?

What is the sum of the green angles in the image below?

Definition: The interior angles of a polygon are all of the angles on the inside of a polygon formed by each pair of adjacent sides.

This quiz will focus on a method for calculating the sum of all of the internal angles inside a polygon. We’ll start with quadrilaterals:

Which sum is greater, the sum of the internal angles of the rectangle or the parallelogram pictured below?

This strange quadrilateral is called a dart. One of its angles is greater than $$180^\circ$$.

If $$X^\circ$$ is the sum of the four internal angles of the dart, what can we say about $$X?$$

True or False?

The sum of the internal angles of any quadrilateral is $$360^ \circ.$$

HINT: Here are is a potential proof. Read and see if you trust it.

Proof 1. Split it into triangles!

In any quadrilateral, you can draw a line in between two opposite corners splitting the quadrilateral into two triangles. Since each triangle has an internal angle sum of $$180^ \circ,$$ the quadrilateral must have a total internal angle sum of $$360^ \circ.$$

Now extend the same idea (and proof techniques if you want) to polygons with more sides!

Which is greater, the sum of all of the red angles below or the sum of all of the blue angles? (Note some of the red and blue angles are overlapping.)

Which is greater, the sum of the irregular pentagon's five orange angles or the sum of the irregular hexagon’s six blue angles? (Note that some of the orange angles and blue angles are overlapping.)

Conclusion: The dissection of any polygon into triangles allows us to calculate the total sum of its internal angles. As a result, we can conclude that

• any quadrilateral has 360 total internal degrees
• any pentagon has 540 total internal degrees
• any hexagon has 720 total internal degrees
• any heptagon has $$\text{ ______ }$$ total internal degrees.

How many total internal degrees are there in a heptagon?
Fill in the blank.

Hint:

The table below shows the sum of the interior angles for many different polygons.

Which of the given options is the general formula for the sum of the interior angles in a polygon?

The following octagon has 4 right angles. What is the sum of the orange angles?

What is the sum of all of the red angles in the figure below?

Four congruent isosceles triangles overlap. What is the measure of the red angle?

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