Transcript Slide 1

Objectives
• Define polygon, concave / convex
polygon, and regular polygon
• Find the sum of the measures of interior
angles of a polygon
• Find the sum of the measures of exterior
angles of a polygon
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Definition of polygon
• A polygon is a closed plane figure formed
by 3 or more sides that are line segments;
– the segments only intersect at endpoints
– no adjacent sides are collinear
• Polygons are named using letters of
consecutive vertices
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Concave and Convex Polygons
• A convex polygon has no
diagonal with points outside
the polygon
• A concave polygon has at
least one diagonal with points
outside the polygon
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Regular Polygon Definition
• An equilateral polygon has all sides
congruent
• An equiangular polygon has all angles
congruent
• A regular polygon is both equilateral and
equiangular
Note: A regular polygon is always convex
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Sum of Interior Angles in Polygons
Convex Polygon # of
Sides
# of Triangles
from 1 Vertex
Sum of Interior Angle
Measures
Triangle
3
1
1* 180 = 180
Quadrilateral
4
2
2* 180 = 360
Pentagon
5
3
3* 180 = 540
Hexagon
6
4
4* 180 = 720
Heptagon
7
5
5* 180 = 900
Octagon
8
6
6* 180 = 1080
n-gon
n
n–2
(n – 2) * 180
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Example: Sum of Interior Angles
Find m∠ X
Solution: The sum of the measures of
the interior angles for a quadrilateral
is (4 – 2) * 180 = 360
The marks in the illustration indicate that
m∠X = m∠Y. So the sum of all four interior angles is
m∠X + m∠X + 100 + 90 = 360
2 m∠X + 190 = 360
2 m∠X
= 170
m∠X = 85
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Polygon Exterior Angle Sum Theorem
• The sum of the measures of the exterior
angles of a polygon, one at each vertex is
360.
m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360
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Example: Exterior Angle Sum
What is the measure of an interior angle of a
regular octagon?
Solution:
8 * exterior angle = 360 (Ext. Angle Sum)
exterior angle = 45
interior angle = 180 – exterior angle
interior angle = 180 – 45 = 135
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