6.1 Polygons

Goal
Identify and classify
polygons. Find angle
measures of quadrilaterals.
Key Words
• polygon
• side of a polygon
• vertex of a polygon
• diagonal of a polygon
Student Help
VOCABULARY TIP
A side connects
consecutive vertices.
A diagonal connects
nonconsecutive
vertices.
6.1 Polygons
Each traffic sign below is an example of a polygon. Notice that each
sign is formed with straight lines.
A polygon is a plane figure that is formed
by three or more segments called sides . Each
side intersects exactly two other sides at each
of its endpoints. Each endpoint is a vertex
of the polygon.
Two vertices that are the endpoints of the same
side are called consecutive vertices. For example,
in polygon PQRST, R and S are consecutive vertices.
A segment that joins two nonconsecutive
vertices of a polygon is called a diagonal .
Polygon PQRST has two diagonals from
vertex R, RP &* and RT&*.
EXAMPLE 1 Identify Polygons
Is the figure a polygon? Explain your reasoning.
a. b. c. d.
Solution
a. Yes. The figure is a polygon formed by four straight sides.
b. No. The figure is not a polygon because it has a side that is not a
segment.
c. No. The figure is not a polygon because two of the sides intersect
only one other side.
d. Yes. The figure is a polygon formed by six straight sides.
P
P
P
diagonals
T
P
T
R
vertex
side
vertex
S
R
S
6.1 Polygons 303

Student Help
STUDY TIP
To classify a polygon
not listed in the table,
use the number of
sides. For example, a
polygon with 14
sides is a 14-gon. A
polygon with n sides
is an n-gon.
304 Chapter 6 Quadrilaterals
Classifying Polygons You can classify polygons by the number of
sides they have. Some special types of polygons are listed below.
TYPES OF POLYGONS
Triangle
Quadrilateral
Pentagon
EXAMPLE 2 Classify Polygons
Decide whether the figure is a polygon. If so, tell what type. If not,
explain why.
a. b. c. d.
Solution
3 sides
4 sides
5 sides
Hexagon
Heptagon
Octagon
6 sides
7 sides
8 sides
a. The figure is a polygon with four sides, so it is a quadrilateral.
b. The figure is not a polygon because it has some sides that are not
segments.
c. The figure is a polygon with five sides, so it is a pentagon.
d. The figure is not a polygon because some of the sides intersect more
than two other sides.
Identify and Classify Polygons
Decide whether the figure is a polygon. If so, tell what type. If not,
explain why.
1. 2. 3. 4.

Student Help
STUDY TIP
Name a polygon by
listing its vertices
consecutively in either
direction. Two names
for the quadrilateral in
Example 3 are PQRS
and RQPS.
Quadrilaterals A diagonal of a quadrilateral divides it into two triangles,
each with angle measures that add up to 180. So, the sum of the
measures of the interior angles of a quadrilateral is 2 180, or 360.
B
2
A
1
EXAMPLE 3 Find Angle Measures of Quadrilaterals
Find the measure of aS.
P
80
Solution
P
70
70
R
S
Use the fact that the sum of the measures of the interior angles
of a quadrilateral is 360.
maP maQ maR maS 360
Quadrilateral Interior Angles
Theorem
708070maS 360 Substitute angle measures.
220maS 360 Simplify.
ANSWER The measure of aS is 140.
Find Angle Measures of Quadrilaterals
Find the measure of aA.
6
THEOREM 6.1
3
ma1 ma2 ma3 180
B
C 2
C C
3
4
4
Quadrilateral Interior Angles Theorem
5
Words The sum of the measures of the interior
angles of a quadrilateral is 360.
Symbols ma1 ma2 ma3 ma4 360
maS 140 Subtract 220 from each side.
5. C 6. C
7.
B
110
60
55
A
100
D
B
80
160
D
A
D
1
A
6 5
A D
ma4 ma5 ma6 180
1
B
2
72
4
108
C
108
D
A
6.1 Polygons 305
3

6.1
Exercises
Guided Practice
Vocabulary Check
306 Chapter 6 Quadrilaterals
1. What type of polygon has 8 sides? 15 sides?
2. Use the diagram of the pentagon
shown at the right. Name all of
the diagonals from vertex D.
Is the figure a polygon? Explain your reasoning.
3. 4. 5.
Find the measure of aA.
Practice and Applications
Extra Practice
See p. 685.
Skill Check
Homework Help
Example 1: Exs. 8–10
Example 2: Exs. 8–10,
21, 24–27
Example 3: Exs. 15–20,
28
6. A
7.
125
B
70
60
D C
Classifying Polygons Decide whether the figure is a polygon. If so,
tell what type. If not, explain why.
8. 9. 10.
11. Logical Reasoning What is the fewest number of sides a polygon
can have? Explain your answer, then name the polygon.
Visualize It!
12. Two different pentagons
Sketch the figure(s) described.
13. A hexagon with three diagonals drawn from a single vertex
14. A quadrilateral with two obtuse angles
A
A
B
B
105
E
C
D
113
75
C
D

IStudent Help
I CLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Exs. 18–20 is
at classzone.com
Plants
CARAMBOLA, or star fruit,
has a cross section shaped
like a five-pointed star.
Finding Angle Measures Find the measure of aA.
15. A
16. B
17. D
A 95
D
124
110
C
55
Using Algebra Find the value of x.
18. 19. 20.
65
60
80
150
110
(x 30)
B
Parachutes Some gym classes play games using parachutes that look
like the polygon below.
21. Tell how many sides the polygon has
and what type of polygon it is.
22. Polygon LMNPQRST is one name for
the polygon. State two other names
using the vertices.
23. Name all of the diagonals that have
vertex M as an endpoint. Not all of the
diagonals are shown.
Plants Use the following information.
Cross sections of roots and stems often resemble polygons. Next to
each cross section is the polygon it resembles. Tell how many sides
each polygon has and tell what type of polygon it is.
Source: The History and Folklore of North American Wildflowers
24. Virginia Snakeroot 25. Caraway
26. Fennel 27. Poison Hemlock
D
87 63
3x
C
T
S
C 100
100
2x
84
L M
R
P
95
B
2x
N
P
A
6.1 Polygons 307

Standardized Test
Practice
Mixed Review
Algebra Skills
308 Chapter 6 Quadrilaterals
28. Technology Use geometry
software to draw a quadrilateral.
Measure each interior angle and
calculate the sum. What happens
to the sum as you drag the
vertices of the quadrilateral?
29. Multi-Step Problem Envelope manufacturers fold a speciallyshaped
piece of paper to make an envelope, as shown below.
●1
a. How many sides are formed by the outer edges of the paper
before it is folded? Name the type of polygon.
b. Tell how many sides are formed by the outer edges of the paper
in Steps 2–4. Name the type of polygon formed after each step.
c. If the four angles of the red quadrilateral in Step 4 are
congruent, then what is the measure of each angle?
Line Relationships Determine whether the lines are parallel,
perpendicular, or neither. (Lesson 3.1)
30. AB ^&*( and CE ^&*( 31. AC ^&*( and BE ^&*(
32. AB ^&*( and AC ^&*( 33. AC ^&*( and CE ^&*(
Finding Angle Measures Find the measure of the numbered angle.
(Lesson 3.4)
34. 35. 36.
2
108
1
●2 ●3 ●4
Distributive Property Use the distributive property to rewrite the
expression without parentheses. (Skills Review, p. 671)
37. 4(x 3) 38. (x 1)6 39. 2(x 7)
40. 5(2x 3) 41. 3(5x 2) 42. (4x 4)x
72
D
A
A
C
3
B
B
C
E
97