Properties and Attributes of Polygons

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6-1
Properties and Attributes of Polygons
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up 1. A ? is a three-sided
polygon. 2. A ? is a four-sided
polygon. Evaluate each expression for n 6. 3.
(n 4) 12 4. (n 3) 90 Solve for a. 5. 12a
4a 9a 100
triangle
quadrilateral
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270
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Objectives
Classify polygons based on their sides and
angles. Find and use the measures of interior
and exterior angles of polygons.
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Vocabulary
side of a polygon vertex of a polygon diagonal reg
ular polygon concave convex
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In Lesson 2-4, you learned the definition of a
polygon. Now you will learn about the parts of a
polygon and about ways to classify polygons.
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Each segment that forms a polygon is a side of
the polygon. The common endpoint of two sides is
a vertex of the polygon. A segment that connects
any two nonconsecutive vertices is a diagonal.
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You can name a polygon by the number of its
sides. The table shows the names of some common
polygons.
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Example 1A Identifying Polygons
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of sides.
polygon, hexagon
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Example 1B Identifying Polygons
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of sides.
polygon, heptagon
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Example 1C Identifying Polygons
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of sides.
not a polygon
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Check It Out! Example 1a
Tell whether each figure is a polygon. If it is a
polygon, name it by the number of its sides.
not a polygon
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Check It Out! Example 1b
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of its sides.
polygon, nonagon
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Check It Out! Example 1c
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of its sides.
not a polygon
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All the sides are congruent in an equilateral
polygon. All the angles are congruent in an
equiangular polygon. A regular polygon is one
that is both equilateral and equiangular. If a
polygon is not regular, it is called irregular.
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A polygon is concave if any part of a diagonal
contains points in the exterior of the polygon.
If no diagonal contains points in the exterior,
then the polygon is convex. A regular polygon is
always convex.
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Example 2A Classifying Polygons
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
irregular, convex
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Example 2B Classifying Polygons
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
irregular, concave
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Example 2C Classifying Polygons
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
regular, convex
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Check It Out! Example 2a
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
regular, convex
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Check It Out! Example 2b
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
irregular, concave
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To find the sum of the interior angle measures of
a convex polygon, draw all possible diagonals
from one vertex of the polygon. This creates a
set of triangles. The sum of the angle measures
of all the triangles equals the sum of the angle
measures of the polygon.
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In each convex polygon, the number of triangles
formed is two less than the number of sides n. So
the sum of the angle measures of all these
triangles is (n 2)180.
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Example 3A Finding Interior Angle Measures and
Sums in Polygons
Find the sum of the interior angle measures of a
convex heptagon.
(n 2)180
Polygon ? Sum Thm.
(7 2)180
A heptagon has 7 sides, so substitute 7 for n.
900
Simplify.
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Example 3B Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each interior angle of a
regular 16-gon.
Step 1 Find the sum of the interior angle
measures.
(n 2)180
Polygon ? Sum Thm.
Substitute 16 for n and simplify.
(16 2)180 2520
Step 2 Find the measure of one interior angle.
The int. ?s are ?, so divide by 16.
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Example 3C Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each interior angle of
pentagon ABCDE.
Polygon ? Sum Thm.
(5 2)180 540
Polygon ? Sum Thm.
m?A m?B m?C m?D m?E 540
35c 18c 32c 32c 18c 540
Substitute.
135c 540
Combine like terms.
c 4
Divide both sides by 135.
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Example 3C Continued
m?A 35(4) 140
m?B m?E 18(4) 72
m?C m?D 32(4) 128
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Check It Out! Example 3a
Find the sum of the interior angle measures of a
convex 15-gon.
(n 2)180
Polygon ? Sum Thm.
(15 2)180
A 15-gon has 15 sides, so substitute 15 for n.
2340
Simplify.
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Check It Out! Example 3b
Find the measure of each interior angle of a
regular decagon.
Step 1 Find the sum of the interior angle
measures.
(n 2)180
Polygon ? Sum Thm.
Substitute 10 for n and simplify.
(10 2)180 1440
Step 2 Find the measure of one interior angle.
The int. ?s are ?, so divide by 10.
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In the polygons below, an exterior angle has been
measured at each vertex. Notice that in each
case, the sum of the exterior angle measures is
360.
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Example 4A Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each exterior angle of a
regular 20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. ?s 360.
Polygon ? Sum Thm.
A regular 20-gon has 20 ? ext. ?s, so divide the
sum by 20.
The measure of each exterior angle of a regular
20-gon is 18.
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Example 4B Finding Interior Angle Measures and
Sums in Polygons
Find the value of b in polygon FGHJKL.
Polygon Ext. ? Sum Thm.
15b 18b 33b 16b 10b 28b 360
120b 360
Combine like terms.
b 3
Divide both sides by 120.
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Check It Out! Example 4a
Find the measure of each exterior angle of a
regular dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. ?s 360.
Polygon ? Sum Thm.
A regular dodecagon has 12 ? ext. ?s, so divide
the sum by 12.
The measure of each exterior angle of a regular
dodecagon is 30.
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Check It Out! Example 4b
Find the value of r in polygon JKLM.
4r 7r 5r 8r 360
Polygon Ext. ? Sum Thm.
24r 360
Combine like terms.
r 15
Divide both sides by 24.
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Example 5 Art Application
Ann is making paper stars for party decorations.
What is the measure of ?1?
?1 is an exterior angle of a regular pentagon. By
the Polygon Exterior Angle Sum Theorem, the sum
of the exterior angles measures is 360.
A regular pentagon has 5 ? ext. ?, so divide the
sum by 5.
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Check It Out! Example 5
What if? Suppose the shutter were formed by 8
blades instead of 10 blades. What would the
measure of each exterior angle be?
?CBD is an exterior angle of a regular octagon.
By the Polygon Exterior Angle Sum Theorem, the
sum of the exterior angles measures is 360.
A regular octagon has 8 ? ext. ?, so divide the
sum by 8.
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Lesson Quiz
1. Name the polygon by the number of its sides.
Then tell whether the polygon is regular or
irregular, concave or convex. 2. Find the sum of
the interior angle measures of a convex 11-gon.
nonagon irregular concave
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3. Find the measure of each interior angle of a
regular 18-gon. 4. Find the measure of each
exterior angle of a regular 15-gon.
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