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72397_116NA_Geom_Polygons
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Table of Contents
To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Session I: Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
NOTICE: Photocopying any part of this book is forbidden by law.
Session II: Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
Lesson 1:
Identify and Use Polygon Properties . . . . . . . . . . . . . . . . . . . . . . . .22
A. Define Terms Related to Polygonal Relationships . . . . . . . . . . . . . .22
B. Classify Two-Dimensional Geometric Figures . . . . . . . . . . . . . . . . .26
C. Use Congruence of Polygons to Solve Problems . . . . . . . . . . . . . .31
D. Use Sums of Interior Angles of Triangles to Find Sums
of Angles of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
E. Use Characteristics of Polygons to Solve Problems . . . . . . . . . . . .37
Lesson 2:
Identify and Use Quadrilateral Properties . . . . . . .
A. Classify Special Quadrilaterals by Characteristics
B. Find Missing Measures of Special Quadrilaterals .
C. Describe Properties of Quadrilateral Relationships
Lesson 3:
Use Coordinate Geometry . . . . . . . . . . . . . . . .
A. Use the Distance Formula to Find Measures
and Classify Quadrilaterals . . . . . . . . . . . . . .
B. Know How to Calculate the Slope of a Line . .
C. Know That Slopes of Parallel Lines Are Equal
D. Know that Slopes of Perpendicular Lines Are
Opposite Reciprocals . . . . . . . . . . . . . . . . . .
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Session III: Posttest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82
Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88
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Identify and Use Polygon
Properties
Define Terms Related to Polygonal
Relationships
A polygon is a closed two-dimensional geometric figure formed by three or more line
segments that meet at their endpoints. These line segments are called sides. Each point
where two sides of a polygon meet is called a vertex. A polygon always has as many
sides as it has vertices (plural of vertex). The figure below shows a polygon that has six
sides and six vertices.
side
vertex
A line segment connecting two nonconsecutive vertices of a polygon is called a diagonal.
The dotted segments in the polygons below are diagonals.
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Concave
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Convex
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Lesson 1: Identify and Use Polygon Proper ties
Look at the polygon on the left of the previous page. All the diagonals of that polygon
lie inside the polygon. In fact, any line segment drawn between any two points of that
polygon would lie inside the polygon. That is because the polygon on the left is convex.
Notice that the polygon on the right looks a little “caved in.” That is because it is a
concave polygon. In a concave polygon, it is possible to draw a line segment between
two points of the polygon that lies partially or entirely outside the polygon.
Most of the important facts and properties about polygons apply to convex polygons. So,
in this book, polygon will mean convex polygon unless otherwise noted.
You should also know about the angles of a polygon. An angle that is formed when two
sides of a polygon meet at a vertex is called an interior angle of the polygon, or simply
an angle of the polygon. An angle formed at the point where a side and an extension of an
adjacent side meet is called an exterior angle.
interior angle
exterior angle
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If all the sides of a polygon are equal in length, the polygon is equilateral. If all of the
interior angles have equal measures, the polygon is equiangular. If a polygon’s sides are
all equal in length and its angles are also all equal in measure, then the polygon is called a
regular polygon.
Equilateral
Equiangular
Regular
(Equilateral and Equiangular)
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Skills Coach, Geometr y: Streamline to Proficiency—Polygons
Coached Practice
Use all appropriate vocabulary words
from the lesson to describe each of
the following:
1
A
figure ABCD, C
D
, A
C
, point D, D, 1, DAB
D
B
C
STRATEGY:
Review the meanings of the bold vocabulary words in the lesson, and use
what you know about geometry symbols. For example, C
D
refers to line
segment CD.
SOLUTION:
Figure ABCD is a polygon. All four of its sides are equal in length, so
figure ABCD is an equilateral polygon. Since any line segment drawn
between any two points inside this polygon will lie inside the polygon,
figure ABCD is convex.
C
D
is a side of the polygon.
A
C
is a diagonal of polygon ABCD because it connects vertices A and C,
which are not consecutive.
Point D is a vertex of the polygon.
1 is an exterior angle of the polygon because it is formed by side AB
and an extension of the adjacent side B
C
.
DAB is an interior angle of polygon ABCD because it is formed by
sides AD
and AB
. Three letters are used to name DAB because there is
more than one angle with vertex A.
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D is an interior angle of the polygon because it is formed by sides DA
and DC
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Lesson 1: Identify and Use Polygon Proper ties
Independent Practice
1.
a.
List all the vocabulary words from the lesson
that describe figure PQRST.
P
T
b.
Q
What terms can you use to describe
point S? QR
? TPQ?
S
2.
Look at polygon ABCDE.
a.
R
A
Name each exterior angle shown, using
three letters to name each.
C
E
b.
L
B
How many diagonals can you draw from
vertex A? Name them.
NOTICE: Photocopying any part of this book is forbidden by law.
D
M
3.
Is it possible for a polygon to be regular if it is not equilateral? Explain your answer.
4.
A triangle contains a 75 angle. Is the triangle equiangular? Explain your answer.
5.
Will two diagonals of a convex polygon always intersect? Make a sketch to illustrate your
answer.
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Skills Coach, Geometr y: Streamline to Proficiency—Polygons
B
Classify Two-Dimensional
Geometric Figures
The table at the right shows how some
polygons are classified based on the
number of sides. Recall that in any
polygon, the number of sides equals
the number of vertices, so you could
also classify a polygon by counting
its vertices.
Number of sides
Classification
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
In addition, polygons can be classified by the lengths of their sides. A polygon with all
sides congruent, or equal in length, is equilateral.
Triangles can be further classified as shown below.
Equilateral
(all sides congruent)
Isosceles
(at least two sides congruent)
Scalene
(no sides congruent)
Triangles can be classified by the measures of their angles. Recall that an acute angle
measures less than 90, a right angle measures exactly 90, and an obtuse angle measures
greater than 90 but less than 180. The examples below show how triangles can be
classified based on the types of angles they have.
Equiangular
(all angles congruent)
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Acute
(all angles acute)
Right
(one right angle)
Obtuse
(one obtuse angle)
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Another way to classify polygons is by angle measure. A polygon with all angles
congruent, or equal in measure, is equiangular.
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Lesson 1: Identify and Use Polygon Proper ties
Remember, if a polygon is both equilateral and equiangular, then it is a regular polygon.
A regular polygon that is also a triangle is called an equilateral triangle. An equilateral
triangle has three congruent sides and three congruent 60 angles.
When classifying polygons, it is also helpful to remember that a polygon can be classified
as convex or concave. Remember, if a line segment drawn between two points of a
polygon lies partially or entirely outside the polygon, it is concave. If no such line segment
can be drawn, it is convex.
Coached Practice
How can you classify each polygon?
1.
STRATEGY:
2.
3.
4.
Here are some of the questions you can ask yourself:
How many sides does the polygon have?
Is it equilateral? equiangular? regular?
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If it is a triangle, what special names apply?
Is it concave?
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Skills Coach, Geometr y: Streamline to Proficiency—Polygons
SOLUTION:
1. This is a pentagon because it has five sides.
It is concave because a line segment connecting
two of its points can be drawn outside the
pentagon as shown by this diagram.
2. The figure for question 2 is an isosceles triangle because it has exactly two congruent
sides. It is an acute triangle because all three angles are acute. So, it is an isosceles
acute triangle.
3. This is a scalene triangle because no sides are congruent. It is obtuse because it has
one angle that is greater than 90. So, it is a scalene obtuse triangle.
4. This is an octagon because it has 8 sides. It is equiangular because all its interior
angles are congruent. So, it is an equiangular octagon. It is not a regular octagon
because its sides do not appear to have equal lengths.
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Lesson 1: Identify and Use Polygon Proper ties
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Independent Practice
1.
How can you classify this polygon?
2.
How can you classify this polygon?
3.
What are all the classifications that apply to both of these polygons?
60º
60º
60º
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Skills Coach, Geometr y: Streamline to Proficiency—Polygons
4.
What is the difference between these two polygons?
5.
Draw an example of each polygon described below. Use marks to show congruent angles or
congruent sides. If any of these polygons cannot be drawn, explain why.
a.
an isosceles obtuse triangle
b.
an isosceles scalene triangle
c.
a scalene right triangle
d.
an obtuse right triangle
e.
a pentagon with exactly two sides congruent
f.
a regular quadrilateral
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