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Tambourines The frame of the tambourine shown is a regular heptagon. What is the measure of each angle of the heptagon?

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An interior angle of a polygon is an angle inside the polygon. You can find the measure of an interior angle of a regular polygon by dividing the sum of the measures of the interior angles by the number of sides.

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Measures of Interior Angles of a Convex Polygon

The sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2) • 180˚.

The measure of an interior angle of a regular n-gon is given by

the formula .(n – 2) • 180˚

n

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EXAMPLE 1 Finding the Sum of a Polygon’s Interior Angles

Find the sum of the measures of the interior angles of the polygon.

SOLUTION

For a convex pentagon, n = 5.

(n – 2) • 180˚ = (5 – 2) • 180˚

= 3 • 180˚

= 540˚

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EXAMPLE 1 Finding the Sum of a Polygon’s Interior Angles

Find the sum of the measures of the interior angles of the polygon.

SOLUTION

For a convex octagon, n = 8.

(n – 2) • 180˚ = (8 – 2) • 180˚

= 6 • 180˚

= 1080˚

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Find the measure of an interior angle of the frame of the heptagonal tambourine.

EXAMPLE 2 Finding the Measure of an Interior Angle

SOLUTION

Because the tambourine is a regular heptagon, n = 7.

Measure of aninterior angle =

(n – 2) • 180˚n

=(7 – 2) • 180˚

7

≈ 128.6˚

Write formula.

Substitute 7 for n.

Evaluate. Use a calculator.

ANSWERThe measure of an interior angle of the frame of the tambourine is about 128.6˚.

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Exterior Angles When you extend a side of a polygon, the angle that is adjacent to the interior angle is an exterior angle. In the diagram, 1 and 2 are exterior angles. An interior angle and an exterior angle at the same vertex form a straight angle.

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Find m 1 in the diagram.

EXAMPLE 3 Finding the Measure of an Exterior Angle

SOLUTION

The angle that measures 87˚ forms a straight angle with 1, which is the exterior angle at the same vertex.

m 1 = 93˚

Angles are supplementary.

Subtract 87˚ from each side.

m 1 + 87˚ = 180˚

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Teapots The diagram shows a teapot in the shape of a regular hexagon. Find m 2.

EXAMPLE 4 Finding an Angle Measure of a Regular Polygon

SOLUTION

Angles are supplementary.

Substitute formula for m 1.

m 1 + m 2 = 180˚

The measure of an interior angle of a regular hexagon is .(6 – 2) • 180˚

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+ m 2 = 180˚(6 – 2) • 180˚

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120˚ + m 2 = 180˚

m 2 = 60˚ Subtract 120˚ from each side.

Simplify.

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Sum of Exterior Angle Measures Each vertex of a convex polygon has two exterior angles. If you draw one exterior angle at each vertex, then the sum of the measures of these angles is 360˚. The calculations below show that this is true for a triangle.

m 4 + m 5 + m 6 = (180˚ – m 1) + (180˚ – m 2) + (180˚ – m 3)

= (180˚ + 180˚ + 180˚) – (m 1 + m 2 + m 3)

= 360˚= 540˚ – 180˚

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Find the unknown angle measure in the diagram.

EXAMPLE 5 Using the Sum of Measures of Exterior Angles

SOLUTION

Sum of measures of exterior

angles of convex polygon is 360˚.x˚ + 81˚ + 100˚ + 106˚ = 360˚

x + 287 = 360

x = 73 Subtract 287 from each side.

Add.

ANSWER The angle measure is 73˚.

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