Post on 05-Jan-2016
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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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