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6-1
ofPolygons
Polygons
Properties and
and Attributes
Attributes of
6-1 Properties
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
6-1 Properties and Attributes of Polygons
Warm Up
1. A Triangle
_____
is a three-sided polygon.
2. A quadrilateral
_____ ___ is a four-sided polygon.
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.
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
side of
Each segment that forms a polygon is a ________
the polygon
_____________.
vertex of
The common endpoint of two sides is a ________
the polygon .
_________
A segment that connects any two nonconsecutive
diagonal
vertices is a __________.
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
Example 1A: Identifying Polygons
1. 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
2. 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
3. Tell whether the figure is a polygon. If it is
a polygon, name it by the number of sides.
not a polygon because the figure has a curved side
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 1a
4. Tell whether each figure is a polygon. If it
is a polygon, name it by the number of its
sides.
not a polygon because it’s not a closed figrue
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 1b
5. Tell whether the figure is a polygon. If it is
a polygon, name it by the number of its sides.
polygon, nonagon
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 1c
6. Tell whether the figure is a polygon. If it is
a polygon, name it by the number of its sides.
not a polygon because it has a curved side
Holt Geometry
6-1 Properties and Attributes of Polygons
A regular polygon is one that is both equilateral and
equiangular.
If a polygon is not regular, it is called irregular.
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
7. 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 2C: Classifying Polygons
8. 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
Example 2B: Classifying Polygons
9. 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
Check It Out! Example 2a
10. 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
11. 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
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
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
12. Find the sum of the interior angle measures of
convex heptagon.
(n – 2)180°
Polygon  Sum Thm.
(7 – 2)180°
A heptagon has 7 sides,
so substitute 7 for n.
900°
Holt Geometry
Simplify.
6-1 Properties and Attributes of Polygons
Example 3B: Finding Interior Angle Measures and
Sums in Polygons
13. 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
14. 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
Substitute.
135c = 540 Combine like terms.
c=4
Holt Geometry
Divide both sides by 135.
6-1 Properties and Attributes of Polygons
Example 3C Continued
mA = 35(4°) =
140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 3a
15. 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°
Holt Geometry
Simplify.
6-1 Properties and Attributes of Polygons
Check It Out! Example 3b
16. 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.
(10 – 2)180° = 1440°
Substitute 10 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 10.
Holt Geometry
6-1 Properties and Attributes of Polygons
17. 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.
Holt Geometry
6-1 Properties and Attributes of Polygons
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 4A: Finding Interior Angle Measures and
Sums in Polygons
18. 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°.
measure of one ext.  =
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°.
Holt Geometry
6-1 Properties and Attributes of Polygons
Example 4B: Finding Interior Angle Measures and
Sums in Polygons
19. 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
Holt Geometry
Divide both sides by 120.
6-1 Properties and Attributes of Polygons
Check It Out! Example 4a
20. Find the measure of each exterior angle
of a regular dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360°.
measure of one ext.
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°.
Holt Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 4b
21. Find the value of r in polygon JKLM.
Polygon
Ext.  Sum Thm.
4r° + 7r° + 5r° + 8r° = 360
°
24r = 360
r = 15
Holt Geometry
Combine like terms.
Divide both sides by 24.