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Polygon Exterior Angle Sum Theorem

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Polygon Exterior Angle Sum Theorem

Study Guide

Key Definition

The sum of the measures of the exterior angles of a convex polygon is $360^\circ$.

Important Notes

The exterior angle sum of any polygon is always $360^\circ$.
For a convex polygon, each exterior angle contributes to the total $360^\circ$.
The formula for the exterior angle sum is derived from the linear pair and interior angle sum.
In non-convex polygons, some exterior angles may be considered negative.
Understanding this theorem helps in solving complex geometry problems.

Mathematical Notation

$n$ is the number of sides

$\angle$ represents an angle

$360^\circ$ is the sum of all exterior angles

Remember to use proper notation when solving problems

Why It Works

Each exterior angle at a vertex is supplementary to the interior angle (they form a linear pair). Since the interior angles of an n-sided polygon sum to $(n-2)\times180^\circ$, summing all exterior angles gives $n\times180^\circ - (n-2)\times180^\circ = 360^\circ$. Hence, the total always equals $360^\circ$.

Remember

The sum of the exterior angles of any convex polygon is always $360^\circ$.

Quick Reference

Exterior Angle Sum:$360^\circ$

Understanding Polygon Exterior Angle Sum Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In any convex polygon, as you move around its edges and extend each side, the turn you make at each vertex is an exterior angle, and all these turns together add up to a full rotation of $360^\circ$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of exterior angles of a convex quadrilateral? $W$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a park shaped like a regular pentagon, each corner having an exterior angle. How many degrees is each exterior angle?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a hexagon, find the measure of one exterior angle if all are equal. Use $360^\circ$.

Challenge Quiz

Single Choice Quiz

Advanced

In a non-convex polygon, some exterior angles are negative. How does this affect the sum?

Recap

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Review key concepts and takeaways