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Assignment. P. 510-513: 1, 2-10 even, 11-21, 24-28, 37-41 Inscribed Polygons Worksheet. Example 1. What is the sum of the interior angles in the polygon below?. Example 2. What’s the difference between convex and concave polygons?. Investigation 1. - PowerPoint PPT PresentationTRANSCRIPT
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Assignment• P. 510-513: 1, 2-10
even, 11-21, 24-28, 37-41
• Inscribed Polygons Worksheet
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Example 1What is the sum of the interior angles in the
polygon below?
S
U
M
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Example 2What’s the difference between convex and
concave polygons?
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Investigation 1Using the two previous concepts, we will
discover a method for finding the sum of the angles in any convex n-gon, where n is the number of sides (or angles) of a given polygon.
Step 1: Draw a series of convex n-gons, starting with n = 3 and ending with n = 6.
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Investigation 1
Step 2: In each polygon, draw all of the diagonals from one vertex. Notice how these diagonals divide the polygons into triangles. How could this help find the sum of the angles in each n-gon?
180
180
180180
180
180
180
180
180180
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Investigation 1
Step 3: Complete the table.
180
180
180180
180
180
180
180
180180
Number of sides 3 4 5 6
Number of triangles
Angle Sum
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Investigation 1
Step 4: Find a formula.
180
180
180180
180
180
180
180
180180
Number of sides 3 4 5 6
Number of triangles 1 2 3 4
Angle Sum 180° 360° 540° 720°
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8.1 Find Angle Measures in Polygons
Objectives:1. To find the sum of the measures of the
interior and exterior angles in any n-gon
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Polygon Interior Angles TheoremThe sum of the measures of the interior
angles of a convex n-gon is (n – 2)·180°.
m1 + m2 + … + mn = (n – 2)·180°
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Example 3What is the sum of the measures of the
interior angles of a convex octagon?
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Example 4What is the measure of each angle of an
equiangular octagon?
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Example 5Find the values of e and f.
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Example 6What is the measure of each angle in any
equiangular n-gon?
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Equiangular Polygon TheoremThe measure of each angle of an
equiangular n-gon can be found by using either of the following expressions:
nnn
360180or 180)2(
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Example 7In a regular polygon, the measure of each
angle is 150˚. How many sides does the polygon have?
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Example 8: SATIf the degree measures of the angles of a
quadrilateral are 4x, 7x, 9x, and 10x, what is the sum of the measures of the smallest angle and the largest angle?
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Investigation 2When you extend one
side of a triangle, you form an exterior angle. If you extend each side of a polygon to form one exterior angle at each vertex, you create a set of exterior angles for the polygon.
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Investigation 2Use the GSP activity
to investigate the sum of the measures of a set of exterior angles of a polygon.
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Polygon Exterior Angles TheoremThe sum of the measures of one set of
exterior angles of a polygon is 360°.
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Example 9What is the value of x?
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Example 10What is the number of sides of a polygon in
which the sum of the degree measures of the interior angles is 4 times the sum of the degree measures of the exterior angles?
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Example 11What is the measure of each exterior angle
in an equiangular octagon?
What is the measure of each exterior angle in an equiangular n-gon? How does this relate to the Equiangular Polygon Theorem?
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Assignment• P. 510-513: 1, 2-10
even, 11-21, 24-28, 37-41
• Inscribed Polygons Worksheet