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5.28: Exterior Angles in Convex Polygons

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    Measure of angles on the outside of a polygon formed by extending a side.

    Exterior Angle Sum Theorem

    An exterior angle is an angle that is formed by extending a side of the polygon.

    f-d_93c55f216c91786218045f3400bcd8e12ff0fc8874fea71333700992+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    As you can see, there are two sets of exterior angles for any vertex on a polygon, one going around clockwise (1st hexagon), and the other going around counter-clockwise (2nd hexagon). The angles with the same colors are vertical and congruent.

    The Exterior Angle Sum Theorem states that the sum of the exterior angles of ANY convex polygon is \(360^{\circ}\). If the polygon is regular with n sides, this means that each exterior angle is \(\dfrac{360^{\circ}}{n}\).

    What if you were given a seven-sided regular polygon? How could you determine the measure of each of its exterior angles?

    Example \(\PageIndex{1}\)

    What is the measure of each exterior angle of a regular 12-gon?

    Solution

    Divide \(360^{\circ}\) by the given number of sides.

    \(30^{\circ}\)

    Example \(\PageIndex{2}\)

    What is the measure of each exterior angle of a regular 100-gon?

    Solution

    Divide \(360^{\circ}\) by the given number of sides.

    \(3.6^{\circ}\)

    Example \(\PageIndex{3}\)

    What is y?

    f-d_aa7748309ebb648d121733e41ca0653dc76b0f6fbf857960e96912ae+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Solution

    \(y\) is an exterior angle and all the given angles add up to \(360^{\circ}\). Set up an equation.

    \(\begin{aligned} 70^{\circ}+60^{\circ}+65^{\circ}+40^{\circ}+y&=360^{\circ} \\ y&=125^{\circ} \end{aligned}\)

    Example \(\PageIndex{4}\)

    What is the measure of each exterior angle of a regular heptagon?

    Solution

    Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. \(\dfrac{360^{\circ}}{7}\approx 51.43^{\circ}\)

    Example \(\PageIndex{5}\)

    What is the sum of the exterior angles in a regular 15-gon?

    Solution

    The sum of the exterior angles in any convex polygon, including a regular 15-gon, is \(360^{\circ}\).

    Review

    1. What is the measure of each exterior angle of a regular decagon?
    2. What is the measure of each exterior angle of a regular 30-gon?
    3. What is the sum of the exterior angles of a regular 27-gon?

    Find the measure of the missing variables:

    1. f-d_a1152bf181eba513d57b67d135f8e7c0df339fe680e57f42eb7d0f7d+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{3}\)
    2. f-d_69769e9c68321bcecf0ab8080080b54410313cc40badd27aeee383ba+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{4}\)
    1. The exterior angles of a quadrilateral are \(x^{\circ}\), \(2x^{\circ}\), \(3x^{\circ}\), and \(4x^{\circ}\). What is \(x\)?

    Find the measure of each exterior angle for each regular polygon below:

    1. octagon
    2. nonagon
    3. triangle
    4. pentagon

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 6.2.

    Resources

    Vocabulary

    Term Definition
    exterior angle An angle that is formed by extending a side of the polygon.
    regular polygon A polygon in which all of its sides and all of its angles are congruent.
    Exterior Angle Sum Theorem Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360 degrees.

    Additional Resources

    Interactive Element

    Video: Interior and Exterior Angles of a Polygon

    Activities: Exterior Angles in Convex Polygons Discussion Questions

    Study Aids: Polygons Study Guide

    Real World: Exterior Angles Theorem

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