Download G6-1-Polygons

6-1
ofPolygons
Polygons
Properties and
and Attributes
Attributes of
6-1 Properties
Holt
Geometry
Holt
Geometry
6-1 Properties and Attributes of Polygons
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
Holt Geometry
6-1 Properties and Attributes of Polygons
Objectives
Classify polygons based on their sides
and angles.
Find and use the measures of interior
and exterior angles of polygons.
Holt Geometry
6-1 Properties and Attributes of Polygons
Vocabulary
side of a polygon
vertex of a polygon
diagonal
regular polygon
concave
convex
Holt Geometry
6-1 Properties and Attributes of Polygons
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.
Holt Geometry
6-1 Properties and Attributes of Polygons
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.
Holt Geometry
6-1 Properties and Attributes of Polygons
You can name a polygon
by the number of its
sides. The table shows
the names of some
common polygons.
Holt Geometry
6-1 Properties and Attributes of Polygons
Remember!
A polygon is a closed plane figure formed by
three or more segments that intersect only at
their endpoints.
Holt Geometry
6-1 Properties and Attributes of Polygons
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
Holt Geometry
6-1 Properties and Attributes of Polygons
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
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 1C: Identifying Polygons
Tell whether the figure is a polygon. If it is a
polygon, name it by the number of sides.
Holt Geometry
6-1 Properties and Attributes of Polygons
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.
Holt Geometry
6-1 Properties and Attributes of Polygons
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.
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 2A: Classifying Polygons
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
irregular, convex
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 2B: Classifying Polygons
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
irregular, concave
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 2C: Classifying Polygons
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
regular, convex
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 2a
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
regular, convex
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 2b
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
Holt Geometry
6-1 Properties and Attributes of Polygons
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.
Holt Geometry
6-1 Properties and Attributes of Polygons
Remember!
By the Triangle Sum Theorem, the sum of the
interior angle measures of a triangle is 180°.
Holt Geometry
6-1 Properties and Attributes of Polygons
Holt Geometry
6-1 Properties and Attributes of Polygons
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°.
Holt Geometry
6-1 Properties and Attributes of Polygons
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°
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Simplify.
6-1 Properties and Attributes of Polygons
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.
(16 – 2)180° = 2520°
Substitute 16 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16.
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 3C: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each
interior angle of pentagon
ABCDE.
(5 – 2)180° = 540° Polygon  Sum Thm.
Polygon 
mA + mB + mC + mD + mE = 540° Sum Thm.
35c + 18c + 32c + 32c + 18c = 540
135c = 540
c=4
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Substitute.
Combine like terms.
Divide both sides by 135.
6-1 Properties and Attributes of Polygons
Check It Out! Example 3a
Find the sum of the interior angle measures of
a convex 15-gon.
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6-1 Properties and Attributes of Polygons
Check It Out! Example 3b
Find the measure of each interior angle of a
regular decagon.
1005
Holt Geometry
6-1 Properties and Attributes of Polygons
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°.
Holt Geometry
6-1 Properties and Attributes of Polygons
Remember!
An exterior angle is formed by one side of a
polygon and the extension of a consecutive side.
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6-1 Properties and Attributes of Polygons
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Example 4B: Finding Interior Angle Measures and
Sums in Polygons
Find the value of b in polygon
FGHJKL.
Holt Geometry
6-1 Properties and Attributes of Polygons
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.
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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