Download 6.1 Polygons

6.1
Goal
Identify and classify
polygons. Find angle
measures of quadrilaterals.
Polygons
Each traffic sign below is an example of a polygon. Notice that each
sign is formed with straight lines.
Key Words
• polygon
• side of a polygon
• vertex of a polygon
• diagonal of a polygon
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.
P
R
vertex
P
side
S
T
Student Help
VOCABULARY TIP
A side connects
consecutive vertices.
A diagonal connects
nonconsecutive
vertices.
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
&* and RT
&*.
vertex R, RP
EXAMPLE
1
P
vertex
R
P
diagonals
T
S
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.
6.1
Polygons
303
Classifying Polygons You can classify polygons by the number of
sides they have. Some special types of polygons are listed below.
Student Help
TYPES OF POLYGONS
STUDY TIP
Triangle
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.
Hexagon
3 sides
6 sides
Quadrilateral
Heptagon
4 sides
7 sides
Pentagon
Octagon
5 sides
EXAMPLE
2
8 sides
Classify Polygons
Decide whether the figure is a polygon. If so, tell what type. If not,
explain why.
a.
b.
c.
d.
Solution
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.
304
Chapter 6
Quadrilaterals
2.
3.
4.
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.
ma1 ⴙ ma2 ⴙ ma3 ⴝ 180ⴗ
B
B
2
3
1
2
C
1
5
6
A
D
C
C
3
4
4
6
5
A
A
D
ma4 ⴙ ma5 ⴙ ma6 ⴝ 180ⴗ
THEOREM 6.1
Quadrilateral Interior Angles Theorem
Words
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.
1
3
4
ma1 ma2 ma3 ma4 360
Symbols
Student Help
2
The sum of the measures of the interior
angles of a quadrilateral is 360.
Find Angle Measures of Quadrilaterals
3
EXAMPLE
Find the measure of aS.
P
80ⴗ
70ⴗ
Solution
R
70ⴗ
P
S
Use the fact that the sum of the measures of the interior angles
of a quadrilateral is 360.
Quadrilateral Interior Angles
Theorem
maP maQ maR maS 360
70 80 70 maS 360
Substitute angle measures.
220 maS 360
Simplify.
maS 140
ANSWER
Subtract 220 from each side.
䊳 The measure of aS is 140.
Find Angle Measures of Quadrilaterals
Find the measure of aA.
5.
C
B
A
6.
60ⴗ
160ⴗ
D
B
80ⴗ
A
72ⴗ
55ⴗ
110ⴗ
100ⴗ
7. B
C
108ⴗ
D
C
A
6.1
108ⴗ
D
Polygons
305
6.1 Exercises
Guided Practice
Vocabulary Check
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.
B
C
A
D
E
Skill Check
Is the figure a polygon? Explain your reasoning.
3.
4.
5.
Find the measure of aA.
A
6.
125ⴗ
B
7.
B
105ⴗ
113ⴗ
D
70ⴗ
60ⴗ
C
A
C
75ⴗ
D
Practice and Applications
Extra Practice
Classifying Polygons Decide whether the figure is a polygon. If so,
tell what type. If not, explain why.
See p. 685.
8.
9.
10.
11. Logical Reasoning What is the fewest number of sides a polygon
can have? Explain your answer, then name the polygon.
Homework Help
Example 1: Exs. 8–10
Example 2: Exs. 8–10,
21, 24–27
Example 3: Exs. 15–20,
28
306
Chapter 6
Visualize It!
Sketch the figure(s) described.
12. Two different pentagons
13. A hexagon with three diagonals drawn from a single vertex
14. A quadrilateral with two obtuse angles
Quadrilaterals
Finding Angle Measures Find the measure of aA.
15.
16.
A
17. D
B
A
95ⴗ
A
D 124ⴗ
C
HOMEWORK HELP
Extra help with problem
solving in Exs. 18–20 is
at classzone.com
55ⴗ
B
D
87ⴗ
63ⴗ
100ⴗ
C
95ⴗ
B
C
Using Algebra Find the value of x.
IStudent Help
ICLASSZONE.COM
110ⴗ
18.
65ⴗ
80ⴗ
19.
60ⴗ
20.
84ⴗ
100ⴗ
150ⴗ
110ⴗ
(x ⴙ 30)ⴗ
3xⴗ
2x ⴗ
2xⴗ
Parachutes Some gym classes play games using parachutes that look
like the polygon below.
21. Tell how many sides the polygon has
L
M
and what type of polygon it is.
22. Polygon LMNPQRST is one name for
the polygon. State two other names
using the vertices.
T
N
S
P
23. Name all of the diagonals that have
vertex M as an endpoint. Not all of the
diagonals are shown.
Plants
R
P
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.
䊳 Source: The History and Folklore of North American Wildflowers
24. Virginia Snakeroot
25. Caraway
26. Fennel
27. Poison Hemlock
CARAMBOLA, or star fruit,
has a cross section shaped
like a five-pointed star.
6.1
Polygons
307
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?
Standardized Test
Practice
B
A
D
C
29. Multi-Step Problem Envelope manufacturers fold a specially-
shaped piece of paper to make an envelope, as shown below.
4
●
3
●
2
●
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?
Mixed Review
Line Relationships Determine whether the lines are parallel,
perpendicular, or neither. (Lesson 3.1)
^&
*( and CE
^&
*(
30. AB
^&
*( and AC
^&
*(
32. AB
^&
*( and BE
^&
*(
31. AC
^&
*( and CE
^&
*(
33. AC
A
B
C
E
Finding Angle Measures Find the measure of the numbered angle.
(Lesson 3.4)
34.
35.
36.
2
108ⴗ
1
Algebra Skills
308
Chapter 6
3
97ⴗ
72ⴗ
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
Quadrilaterals