Download properties of polygons

Solve
Problems
Analyze
Reason
Organize
Integrated Math
Concepts
Measure
Model
Compute
Communicate
Integrated Math Concepts
Module 10
Properties of Polygons
Second Edition
National PASS Center
2006
National PASS Center
BOCES Geneseo Migrant Center
27 Lackawanna Avenue
Mount Morris, NY 14510
(585) 658-7960
(585) 658-7969 (fax)
www.migrant.net/pass
Authors:
Justin Allen
Diana Harke
Editor:
Sally Fox
Desk Top Publishing:
Sally Fox
Developed for Project MATEMÁTICA ((Math Achievement Toward Excellence for
Migrant Students And Professional Development for Teachers in Math Instruction
Consortium Arrangement), a Migrant Education Program Consortium Incentive project,
by the National PASS Center under the leadership of the National PASS Coordinating
Committee with funding from Region 20 Education Service Center, San Antonio,
Texas.
Copyright © 2006 by the National PASS Center. All rights reserved. No part of this
book may be reproduced in any form without written permission from the National
PASS Center.
Solve
Problems
Organize
Analyze
Reason
Integrated Math
Concepts
Model
Compute
Measure
Communicate
Integrated Math Concepts
Module 10
Properties of Polygons
Second Edition
National PASS Center
2006
BOCES Geneseo Migrant Center
27 Lackawanna Avenue
Mount Morris NY 14510
Acknowledgements
The materials included in this Integrated Math Concepts course were gathered, in part, from
the National PASS Center’s Algebra I and Geometry courses which were written by Diana
Harke. Ms. Harke currently is an instructor of mathematics at the State University of New
York at Geneseo where she also supervises student teachers. She is a former junior and senior
high school math teacher with experience in the United States and Canada. Ms. Harke’s
courses produced thus far for the National PASS Center (NPC) have been very well received
across the country, increasing the percentage of PASS mathematics courses being utilized
throughout the migrant education network and beyond. It should be noted that two of the
recent National Migrant PASS Students of the Year, Benancio Galvin of Marana, Arizona
(2004) and Yesenia Medina of San Juan, Texas, and Wild Rose, Wisconsin (2006), have
moved ahead toward their dreams of completing their high school graduation requirements
thanks to their success with Ms. Harke’s Algebra I course.
To meet the needs of migrant students requiring a more condensed resource to strengthen their
math skills, the original curriculum materials were adapted, edited, modified, and expanded by
Mr. Justin Allen. Mr. Allen is a certified secondary level math teacher and is currently
pursuing a graduate degree in secondary education at the State University of New York at
Geneseo. He taught middle school math and Algebra in Canandaigua, New York, for three
years and, most recently, high school math in Livonia, New York. Mr. Allen assisted in the
editing of the PASS Algebra II course which was released early in 2006.
Acknowledgement is offered also to Ms. Sally Fox, Coordinator, National PASS Center, for
her commitment to the development of quality curriculum. As with all materials produced by
the NPC, her involvement with Integrated Math Concepts at all levels has played a key role in
the addition of this offering to the growing number of courses available to migrant students and
others seeking to master the necessary skills to become productive members of society.
Robert Lynch, Director
Module 10 – Properties of Polygons
Table of Contents
Page
Introduction
i
Objectives
1
Review
16
Practice Problems
17
Answers to “Try It” Problems
22
Answers to Practice Problems
25
Glossary of Terms
31
Integrated Math Concepts – Introduction
The PASS Concept
PASS (Portable Assisted Study Sequence) is a study program created to help you earn
credit through semi-independent study with the help of a teacher/mentor. Your teacher/mentor
will meet with you on a regular basis to: answer your questions, review and discuss
assignments and progress, and administer tests. You can undertake courses at your own pace
and may begin a course in one location and complete it in another.
Strategy
Mathematics is not meant to be memorized; it is meant to be understood. This course
has been written with that goal in mind.
Mathematics must not be read in the same way that a novel is read. In order to read a
mathematics text most effectively you must pay close attention to the structure of each
expression and to the order that operations are performed. You might think of mathematics as
you would a foreign language. Every symbol in a mathematical expression is meant to
communicate a message in that language; therefore, to understand the language you must
understand the symbolism.
Always read with a pencil and scrap paper in hand. Make notes in the margins of your
book where you have questions and write “what if” variations to problems to discuss with your
teacher/mentor.
i
Course Content
Integrated Math Concepts is divided into ten modules. Each module teaches concepts
and strategies that are essential for establishing a firm foundation in each content area.
The following is a description of the ten modules in Integrated Math Concepts:
Module 1
Real Numbers
Learn to recognize and differentiate between natural numbers, whole numbers, integers,
rational numbers, irrational numbers, and real numbers.
Relate the number line to the collection of real numbers.
Module 2
Sets
Recognize a well-defined set
Learn set notation and terminology
Study some subsets of real numbers – prime and composite numbers
Module 3
Variables and Axioms
Learn
•
why, when, and how to use a variable
•
the definition of an axiom
•
some specific axioms
Module 4
Properties of Real Numbers
Learn the characteristics and uses of the following properties of real numbers:
•
the commutative property
•
the associative property
•
the distributive property
•
identity elements
•
inverses
•
the multiplication property of zero
•
to understand why division by zero is not allowed
•
to introduce the uniqueness and existence properties
ii
Module 5
Fractions
Become comfortable with fractions by
•
understanding their make-up
•
comparing their sizes
Prepare for operations with algebraic fractions
•
by understanding the concepts behind the algorithms
•
by determining if solutions are reasonable
Module 6
Decimals
Become comfortable with decimals and decimal operations
•
by understanding the relative size of decimals
•
by understanding why the algorithms or rules dealing with decimals work
•
by testing answers for reasonableness
Module 7
Order of Operations
Understand why problems need to be performed in a certain order
Evaluate numerical expressions using order of operations
Evaluate variable expressions for specific values
Module 8
Equations
Translate algebraic expressions and equations, as well as consecutive integer questions
Solve:
•
One-step equations
•
Two-step equations
•
Complex equations (combining like terms, use of the distributive property,
variables on both sides)
•
Multi-step equations
Translate algebraic inequalities
Solve and graph solutions to one and two-step inequalities
iii
Module 9
Geometry
Describe points, lines, and planes
Sketch and label points, lines, and planes
Use problem solving to explore points, lines, and planes
Define line segments, rays, and angles
Recognize and examine types of angles
Explore problems using angle properties
Explore line relationships
Module 10
Properties of Polygons
Recognize and classify 2-dimensional shapes –
circles, triangles and quadrilaterals
Find 2-dimensional shapes in the environment
Explore the sum of the measures of the angles of triangles and quadrilaterals
Classify a polygon according to the number of its sides
Count diagonals in polygons
Find the measures of the interior and exterior angles in polygons
Course Organization
Each module begins with a list of the objectives. This is a short list of what you will
learn.
Definitions, theorems, and
ÍA set is a collection of objects.
mathematical properties appear as
strips of paper tacked to the page so that
they may be easily found. Examples are used to illustrate each new concept. These are
followed immediately by “Try It” problems to see if you
understand the concept. You are to write the answers to the “Try
It” problems right in your book and then check your answers with
the detailed solutions farther back in the module.
iv
Many lessons include the following types of inserts.
“Think Back” boxes – denoted with an arrow pointing backwards. These are
reminders of things that you have probably already learned.
“Problem solving tips” – denoted with a light bulb
“Calculator tips” – denoted with a small calculator
“Algorithms” – denoted with a fancy capital A. An algorithm is a rule (or step by
step process) used to solve a specific type of problem.
"Facts” – denoted by a small flashlight
At the end of each module you will be asked to highlight parts of the lesson as a way to
review the terminology and concepts that you just studied. You will also be asked to write
about something that you learned in your own words or list any questions to ask your
teacher/mentor about something that you did not understand. This last step is extremely
important. You should not continue on to the next activity or module until all your questions
have been answered and you are sure that you thoroughly understand the concept you just
finished.
Finally, you will be asked to practice what you have learned. Athletes in every sport
must practice their skills to become better at their sport. The same is true of mathematicians.
In order to become a good mathematician, you must practice what you have learned so that it
becomes easier and easier to solve problems. You should keep a math journal or notebook
where you will do your practice problems. Detailed answers to the practice problems will be
found toward the end of the module just ahead of the glossary section.
v
A glossary / index of the mathematical terms used in this course has been provided at
the end of each module. It contains definitions as a reference to help your understanding of
these specialized mathematical terms.
Unlike other PASS courses, there is no separate Mentor Manual for this course as all of
the answers to practice problems are provided within each module. Should you require
additional support, do not hesitate to ask your mentor or teacher. That is why they are there.
Testing
When you have completed all the exercises and practice problems in a module and you
and your teacher/mentor feel that you have a good grasp of the material, you will take a test
covering what you should have learned in that module.
Test taking tips
1) Make sure all of your questions have been answered and that you feel confident that
you understand the concepts on which you are to be tested.
2) Do not rush.
3) Be neat. Sometimes handwritten numbers or letters are misread.
4) Be organized. Do computations on a separate piece of paper or, if there is room on
your test sheet, in the space provided, so as to keep the flow of the problem clearly
in focus.
5) Check your answers to see
a) if you actually answered the question that was asked, and
b) that the answer is reasonable.
6) Be aware of the particular types of errors that you are prone to make. Arithmetic
mistakes are often repeated if you merely repeat the computations. Use your
calculator to prevent these types of errors and concentrate on
a) choosing the correct operations,
b) following the proper order of operations, and
c) applying valid mathematical techniques.
vi
National PASS Center
Solve
Problems
Analyze
Organize
Integrated
Math Concepts
Reason
Measure
Properties of
Polygons
Model
Compute
Communicate
Objectives:
Recognize and classify 2-dimensional shapes –
circles, triangles and quadrilaterals
Find 2-dimensional shapes in the environment
Explore the sum of the measures of the angles of triangles and quadrilaterals
Classify a polygon according to the number of its sides
Count diagonals in polygons
Find the measures of the interior and exterior angles in polygons
Í Plane geometry is the branch of mathematics that deals with
figures that lie in a plane or flat surface.
Plane geometry is that part of geometry that deals with two-dimensional figures. You are no
doubt already familiar with many of these figures: circles, squares, rectangles, and triangles,
etc.
Module 10 – Properties of Polygons
1
MATEMÁTICA August 2006
Í A circle is the set of all points in a plane at a given
radius
distance (the radius) from a given point (the center).
center
A circle divides the plane into three sections: those points on the circle, those points outside the
circle, and those points inside the circle.
Points on the circle
Points outside the circle
Points inside the circle
Í A polygon is a closed figure bounded by line segments.
1. Which of the following figures represents a circle?
a.
b.
2.
Is a circle a polygon? Explain your answer.
3.
What is the least number of sides that a polygon can have? Explain.
Integrated Math Concepts
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Í A triangle is a polygon having three sides.
Triangles can be classified according to the measure of their angles.
ÍAn acute triangle is a triangle with three acute angles.
Think Back
Acute means less than
90° .
Thus all 3 angles are less
than 90° .
Í A right triangle is a triangle with a right angle.
Think Back
A right triangle contains one angle = 90° .
Í An obtuse triangle is a triangle with an obtuse angle.
Think Back
Obtuse means greater than 90° .
Thus only one angle needs to be greater than 90° .
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Í An equiangular triangle is a triangle all of whose angles are equal.
The curved marks inside the equiangular triangle indicate that the angles measure the same.
You will discover the angle measurements as a “Try It”.
Triangles also can be classified according to the measure of their sides.
Í A scalene triangle is a triangle with
no two sides congruent.
Í An isosceles triangle is a triangle with two sides congruent.
Í An equilateral triangle is a triangle all of whose sides are congruent.
The single tick marks on the isosceles and equilateral triangles indicate which sides measure
the same. The scalene triangle has a different number of tick marks on each of its sides
indicating that the sides all have different measurements.
Think Back
Congruent means of the same shape and size.
Integrated Math Concepts
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National PASS Center
1. Draw a large acute triangle on scrap paper using your
straight edge or ruler.
2. Label the angles 1, 2, and 3.
3. Tear the angles off the triangle.
4. Draw a line segment and mark a point on the line.
5. Put the vertices of the three angles together at the point.
What is the sum of the three angles?
6. Now draw a right triangle on scrap paper.
You may use the edge of a sheet of paper to make a right angle.
7. Repeat steps 2 – 5. What is the sum of the three angles?
8. Draw an obtuse triangle on scrap paper.
9. Repeat steps 2 – 5. What is the sum of the three angles?
Make a conjecture about the sum of the angles of any triangle.
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
You should have discovered that the sum of the angles of a triangle is 1800 .
Example 1
7x
Find the measure of all of the angles in this triangle.
Solution
12x + 40
Since the sum of the angles of a triangle is 1800 , then
(12 x + 40 ) + 9 x + 7 x = 180
0
9x
0
28 x + 400 = 1800
28 x = 1400
x = 50
7 x = 350
9 x = 450
12 x + 400 = 1000
The angles are 350 , 450 , and 1000 .
4.
How many degrees is each angle of an equiangular triangle? Justify your answer.
5.
If a right triangle also has an angle of 600 , how large is the third angle? Justify
your answer.
C
2x − 40
6.
Find the measure of all the angles in the following triangle.
x
A
Integrated Math Concepts
6
55
B
National PASS Center
ÍA quadrilateral is a polygon
with four sides.
Quadrilaterals can be divided into subcategories by using their characteristics.
Í A parallelogram is a quadrilateral whose opposite sides are parallel
Í A rhombus is a parallelogram with all sides congruent.
Í A rectangle is a parallelogram with all angles congruent.
Í A square is a parallelogram with the same characteristics as the
rhombus and rectangle, where all sides and all angles are congruent.
Í A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Í An isosceles trapezoid is a trapezoid with one pair of
adjacent angles congruent.
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Every parallelogram is by definition a quadrilateral, but not every quadrilateral is a
parallelogram. Every rectangle is by definition a parallelogram, but not every parallelogram is
a rectangle.
Example 2
Name two quadrilaterals other than a rectangle that are parallelograms.
Solution
Squares and rhombi are also parallelograms.
Name a quadrilateral that is not a parallelogram.
Solution
A trapezoid is a quadrilateral that is not a parallelogram.
7. Place the following terms on the tree diagram. Let the more general terms be
placed above the more specific terms. If a branch of the tree connects two terms,
the lower term must be an example of the higher term. The term “triangle” has
been placed for you.
Terms to use: polygon, isosceles triangle, scalene triangle, equilateral
triangle, square, rectangle, rhombus, quadrilateral, parallelogram, isosceles
trapezoid, and trapezoid.
triangle
Integrated Math Concepts
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National PASS Center
ÍThe line segments forming a polygon are called the
Vertex
sides of the polygon. A point where two sides of a
polygon meet is called a vertex of the polygon. A
Side
polygon
polygon divides a plane into three regions: the
interior of the polygon, the exterior of the polygon,
Interior
and the polygon itself.
exterior
Polygons are classified according to the number of sides or vertices that they have. A polygon
has the same number of sides as it does vertices.
triangle
Number of Sides or
Vertices
3
quadrilateral
4
pentagon
5
hexagon
6
heptagon
7
octagon
8
nonagon
9
decagon
10
n-gon
n
Polygon Name
Í
A diagonal is a line segment joining two non-adjacent
vertices of a polygon.
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Example 3
How many diagonals does a triangle have?
Solution
A triangle has no diagonals because there are no non-adjacent vertices.
The diagonals from a single vertex of a polygon (with more than three sides) divide a polygon
into triangles.
Example 4
How many triangles are formed if diagonals are drawn from a single vertex of a quadrilateral?
Solution
Two triangles are formed.
How many triangles are formed if diagonals are drawn from a single vertex of a pentagon?
Solution
Three triangles are formed.
Integrated Math Concepts
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National PASS Center
Determine the number of triangles each polygon is divided
into by drawing diagonals from a single vertex. Let n be
greater than 3.
Number of sides on polygon
Number of triangles
6
7
8
9
10
11
n
Í An interior angle of a polygon is an angle that lies inside the
polygon and is formed by two adjacent sides of the polygon.
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Example 5
What is the measure of the sum of the interior angles in a pentagon?
Solution
Since a pentagon can be divided into three triangles by drawing
diagonals from one of its vertices, the sum of the angles in the
pentagon is the same as the sum of the angles in the three triangles.
Since each triangle has a measure of 1800 , the sum of the measures of the angles in the
pentagon is 3 (1800 ) = 5400 .
1. Sketch a hexagon. Draw all the diagonals from
one vertex. What is the measure of the sum of the
interior angles in a hexagon?
2. Sketch an octagon. Draw all the diagonals from one vertex. What is the measure of
the sum of the interior angles in an octagon?
Integrated Math Concepts
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National PASS Center
3. Sketch a decagon. Draw all the diagonals from one vertex. What is the measure of
the sum of the interior angles in a decagon?
4. Complete the following chart.
No. of sides in polygon
No. of diagonals from a vertex
Sum of interior angles
4
5
6
8
10
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
C
B
Í An exterior angle of a polygon is an angle formed by one
side of the polygon and an adjacent side extended.
A
D
E
Example 6
Find the measure of ∠x and the measure of the
B
C
exterior angle ∠CDE if the measure of the other
angles are as follows:
x
∠A = x + 9 , ∠B = 2 x + 14 , and ∠C = 3x − 27
0
0
0
A
E
D
Solution
Since the sum of the measure of the interior angles of the trapezoid is 3600 ,
x + ( 3 x − 27 0 ) + ( 2 x + 140 ) + ( x + 90 ) = 3600
Since ∠ADE is a straight angle,
7 x − 40 = 3600
∠CDE = 1800 − 520 = 1280
7 x = 3640
x = 520
A
8. Find the measure of the exterior angle y if
∠A = 1080 and ∠F = 1160 .
x - 19
x-8
x
x
Integrated Math Concepts
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y
x +3
F
National PASS Center
Í A regular polygon is a polygon whose sides are all
equal and whose angles are all equal.
Example 7
Find the measure of an interior and an exterior angle in a regular hexagon.
Solution
Since a hexagon can be divided into 4 triangles, the sum of the interior angles is
4 (1800 ) = 7200 . Since a regular hexagon has 6 equal angles, each interior angle is
7200
= 1200 . Since exterior angles are supplementary to interior angles, each exterior
6
angle is 1800 − 1200 = 600 .
9. Find the measure of an interior and an exterior angle in a regular pentagon.
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Review
1. Highlight the following words and their definitions:
a. plane geometry
b. circle
c. polygon
d. triangle
e. acute triangle
f. right triangle
g. obtuse triangle
h. equiangular triangle
i. scalene triangle
j. isosceles triangle
k. equilateral triangle
l. quadrilateral
m. parallelogram
n. rhombus
o. rectangle
p. square
q. trapezoid
r. isosceles trapezoid
s. sides of a polygon
t. vertex of a polygon
u. diagonal
v. interior angle of a polygon
w. exterior angle of a polygon
x. regular polygon
Integrated Math Concepts
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2.
Write about one new thing that you learned in this module or make a list of questions that
you would like to discuss with your mentor.
Practice Problems
Properties of Polygons
Directions: Write your answers in your math journal.
Label this exercise Properties of Polygons.
Connections/Modeling
1. Which of the polygons in the tree diagram of Try It 7 have all equal sides?
2. Which of these triangles are classified according to the measure of their angles and which
are classified according to the measure of their sides: scalene triangles, equilateral
triangles, equiangular triangles, acute triangles, right triangle, isosceles triangles, obtuse
triangles?
3. Find the measure of the third angle of a triangle if the measures of the other two angles are
370 and 560 .
C
4. Find the measures of all the angles in this triangle if ∠A is half
of a right angle, ∠B = 5 x + 3 , and ∠C = 6x
A
B
5. If the measure of one angle of a triangle is 3 times another and the third angle is 200 larger
than the measure of the smaller of the other two angles, what is the measure of all the
angles of the triangle?
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
6. Which of these statements are true and which are false?
a. A square is a rectangle.
b. A rectangle is a square.
c. A trapezoid is a parallelogram.
d. A parallelogram is a trapezoid.
e. A square is a quadrilateral.
f. A quadrilateral is a square.
g. An equilateral triangle is isosceles.
h. A scalene triangle is equilateral.
7. List and identify as many of the different kinds of triangles and quadrilaterals as you can in
the following figure.
8. A polygon divides a plane into what three regions?
9. How does the number of sides in a polygon compare with the number of its vertices?
10. Find the measures of the interior and exterior angles of each of these regular polygons:
a. Heptagon
b. Octagon
c. Nonagon
d. Decagon
B
C
11. Find the values of all the angles in this
D
sketch if AEF is a straight angle and
A
∠D = x + 8, ∠C = 2 x + 5, ∠B = x + 10, and ∠A = x + 13.
3
2
x
E
F
Integrated Math Concepts
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12. If the figure is a regular hexagon, find the measure
of ∠x . (Hint: What do the angle markings mean?)
x
Explorations
1. To find the sum of the angles of a quadrilateral:
a. Make a sketch of a rectangle. Draw a line from one vertex to another non-adjacent
vertex. This is called a diagonal of the rectangle. It divides the rectangle into two
___________. Make a conjecture concerning the sum of the angles of a rectangle.
b. Make sketches of the following quadrilaterals: a square, a trapezoid, a rhombus, and a
general parallelogram. Draw a diagonal in each figure. How many triangles are
formed in each figure? Make a conjecture concerning the sum of the angles in any
quadrilateral.
2. To construct a special angle:
a. Mark a point on your paper and draw a large circle with your compass. Draw a
diameter using your original point. Pick a point on the circle and connect it to the
endpoints of the diameter. Now mark a point on the circle on the other side of the
diameter and connect it to the ends of the diameter. Cut out the two triangles and
compare the size of the angles with their vertex on the circle.
b. Try the experiment again using a circle of a different size. Compare the angles you get
with the angles from the first circle. What did you find?
c. How large do you think these angles are? How can you tell for sure?
d. How does this knowledge help you?
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
3. a. Complete the following chart for the regular polygons with the given number of sides.
Number of sides of
regular polygon
Number of triangles formed
by diagonals from a single
vertex
Sum of the measure
of interior angles
Measure of each
interior angle
20
30
40
50
60
70
80
90
100
200
500
1000
b. Do you think there is a limit to the measure of an interior angle of a polygon? If not,
why not? If so, what is it and why do you think this is so?
c. If sketches of regular polygons are placed side by side with ever increasing number of
sides, the polygons begin to look more and more like _______________.
4. A quadrilateral has two diagonals if diagonals are drawn from every
vertex and not just a single vertex.
a. Sketch a pentagon, hexagon, and an octagon.
b. Draw in all diagonals and count them.
c. Make a conjecture about the number of diagonals that can be drawn altogether in an
n-gon where n is greater than 3.
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Proofs/Justifications/Constructions
1. Is this figure a quadrilateral? Justify your answer.
2. Can a right triangle be obtuse? Justify your answer.
3. Mark two points on your paper (not too close together) and label them points A and B.
Using only your compass and your straight edge (the straight edge is the piece of
equipment in your supplies that looks like a ruler, but has no markings on it), construct a
right angle at point B. Explain why you think your construction is correct.
4. Although it includes a reflex angle, show that the sum of the measures of the interior angles
of the quadrilateral
is 3600 . Justify your answer.
5. The sum of the angles in a regular polygon is 18000 . How many sides does the polygon
have? What is the measure of one of its interior angles? Justify your answer.
6. Suppose a polygon has n sides with n > 3 . Make a conjecture about the sum of the
measures of its angles in terms of the variable n. Justify your answer.
7. Make a conjecture about the measure of each interior angle of a regular n-sided polygon
with n > 3 . Justify your answer.
8. In Exploration #2 on page 11 you made a conjecture as to the number of diagonals that can
be drawn in an n-sided polygon if n > 3 . You may have written your conjecture in this
form: An n-sided polygon has
n ( n − 3)
diagonals. Justify this answer. (Hint: How many
2
diagonals are there from each vertex? Why? How many vertices does an n-sided figure
have? Why is it necessary to divide by 2?)
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
1. Figure b represents the circle. Figure a represents the interior of a circle.
2. A circle is not a polygon since it is not made up of line segments.
3. The least number of sides a polygon can have is three. If you decrease the size of
one side of a triangle until it no longer exists, you end up with two line segments
instead of a polygon.
4. Let x = the measure of one of the angles in the
x + x + x = 1800
equiangular triangle. Since the other angles have
3x = 1800
x = 600
the same measure, each angle is 600 .
5. A right angle has a measure of 900 . Let x = the
measure of the third angle. The angle is 300 .
900 + 600 + x = 1800
1500 + x = 1800
x = 300
2 x − 400 + x + 550 = 1800
6. The sum of the angles is 1800 :
3x + 150 = 1800
3 x = 1650
x = 550
2 x − 400 = 700
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7.
polygon
quadrilateral
triangle
scalene triangle
isosceles triangle
parallelogram
rhombus
trapezoid
rectangle
isosceles trapezoid
equilateral triangle
square
0
0
0
0
0
0
8. 108 + ( x − 8 ) + x + x + ( x + 3 ) + 116 + ( x − 19 ) = 5 (180 )
5 x + 2000 = 9000
5 x = 7000
x = 1400
But the question asks for the measure of y.
y = 1800 − ( x − 190 )
y = 1800 − (1400 − 190 )
y = 590
9. A pentagon can be divided into 3 triangles by drawing diagonals from a single
vertex and since there are 5 vertices in a pentagon, the measure of an interior angle
of a regular pentagon is
3 (1800 )
5
= 1080 . Since an exterior angle is supplementary
to this angle, an exterior angle has a measure of 1800 − 1080 = 720 .
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Explorations
#1. You should have discovered that the sum of all three angles of a triangle is 180° .
#2.
Number of sides on polygon Number of triangles
6
4
7
5
8
6
9
7
10
8
11
9
n
n−2
#3. 1. The measure of the sum of the
interior angles of a hexagon is
4 (1800 ) = 7200 .
2. The measure of the sum of the
interior angles of an octagon is
6 (1800 ) = 10800 .
3. The measure of the sum of the
interior angles of a decagon is
8 (1800 ) = 14400 .
4.
No. of sides in polygon
No. of diagonals from a vertex
Sum of interior angles
4
2
2 (1800 ) = 3600
5
3
3 (1800 ) = 5400
6
4
4 (1800 ) = 7200
8
6
6 (1800 ) = 10800
10
8
8 (1800 ) = 14400
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Answers to Practice Problems
Connections and Modeling
1. The polygons are equilateral triangles, rhombi, and squares.
Classified by the measure of their sides
2. Classified by the measure of their angles
Equiangular triangles
Scalene triangles
Acute triangles
Equilateral triangles
Right triangles
Isosceles triangles
Obtuse triangles
3. Let x = the measure of the third angle.
x + 37 0 + 560 = 1800
x + 930 = 1800
The third angle is 870 .
4. ∠A =
1
2
( 90 ) = 45
0
x = 87 0
0
450 + ( 5 x + 130 ) + 6 x = 1800
11x + 480 = 1800
11x = 1320
x = 120
5 x + 30 = 630
6 x = 720
∠A = 450 , ∠B = 630 , and ∠C = 720
5. Let x = the measure of the smallest angle. Then the measure of the second angle is 3x and
the measure of the third angle is x + 200 .
x + 3x + x + 200 = 1800
5 x = 1600
x = 320
The angles are 320 , 960 , and 520 .
3x = 960
x + 200 = 520
6. a. True
b. False
g. True
h. False
c. False
d. False
e. True
f. False
7. There are 6 triangles – 2 right triangles, 4 other ones (appear to be isosceles). There are 9
quadrilaterals – 6 rectangles, 3 trapezoids (2 appear to be isosceles).
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
8. A polygon divides a plane into these three regions: the interior region, the exterior
region, and the polygon itself.
9. The number of vertices in a polygon is the same as the number of sides it has.
10. a.
An interior angle of a regular heptagon is
5 (1800 )
7
≈ 128.60 .
An exterior angle is approximately 1800 − 128.60 = 51.40 .
b.
An interior angle of a regular octagon is
6 (1800 )
8
= 1350 .
An exterior angle is 1800 − 1350 = 450 .
c.
An interior angle of a regular nonagon is
7 (1800 )
9
= 1400 .
An exterior angle is 1800 − 1400 = 400 .
d.
An interior angle of a regular decagon is
8 (1800 )
10
= 1440 .
An exterior angle is 1800 − 1440 = 360 .
11. Since ∠x and ∠AED are supplementary, ∠AED = 1800 − x .
( x + 13 ) + (
0
3
2
x + 100 ) + ( 2 x + 50 ) + ( x + 80 ) + (1800 − x ) = 3 (1800 )
2 x + 260 + 3 x + 200 + 4 x + 100 + 2 x + 160 + 3600 − 2 x = 2 ( 5400 )
9 x + 4320 = 10800
9 x = 6480
x = 720
∠x = 720
∠A = x + 130 = 720 + 130 = 850
∠B = 32 x + 100 =
3
2
( 72 ) + 10
0
0
= 1180
∠C = 2 x + 50 = 2 ( 720 ) + 50 = 1490
∠D = x + 80 = 720 + 80 = 800
∠AED = 1800 − x = 1800 − 720 = 1080
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12. Each interior angle in a regular hexagon is
4 (1800 )
6
= 1200 .
The marked angles in the triangle are equal.
If y = the measure of one of these angles, then
2 y + 1200 = 1800
2 y = 600
y = 300
Furthermore since x + y = 1200 , x = 1200 − y = 1200 − 300 = 900
Explorations
1. a. Triangles
The sum of the measures of the angles of a rectangle is 3600 .
b. Each figure is divided into two triangles. The sum of the measures of the angles of a
quadrilateral is 3600 .
2. a. The angles have the same measure.
b. The angles have the same measure.
c. The angles are all 900 (right angles). You can compare them to the corner angle on a
piece of paper, use a protractor to measure them, or put two together and see that they
form a straight angle.
d. This provides a way to construct a right angle.
3. a.
Number of
sides of
regular
polygon
Number of
triangles formed
by diagonals from
a single vertex
20
18
18 (1800 ) = 32400
30
28
28 (1800 ) = 50400
40
38
38 (1800 ) = 68400
50
48
48 (1800 ) = 86400
Sum of the measure of
interior angles
Measure of each
interior angle
32400
20
50400
30
68400
40
86400
50
= 1620
= 1680
= 1710
= 172.80
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MATEMÁTICA August 2006
60
58
58 (1800 ) = 104400
70
68
68 (1800 ) = 122400
80
78
78 (1800 ) = 140400
90
88
88 (1800 ) = 158400
100
98
98 (1800 ) = 176400
200
198
198 (1800 ) = 356400
500
498
498 (1800 ) = 896400
1000
998
998 (1800 ) = 1796400
104400
= 1740
60
122400
≈ 174.90
70
140400
= 175.50
80
158400
= 1760
90
176400
= 176.40
100
356400
= 178.20
200
896400
≈ 179.30
500
1796400
≈ 179.60
1000
b. There is a limit. The values are approaching 1800 , but they cannot reach that value
because an interior angle must be less than a straight angle.
c. The polygons will begin to look more like a circle.
4. a.
b.
5 diagonals
9 diagonals
c. Conjecture: An n-gon has
20 diagonals
n ( n − 3)
diagonals.
2
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Justifications / Constructions
1. Yes, the figure is a quadrilateral since it has four sides each of which are line segments.
2. No, a right triangle can’t also be obtuse. An obtuse angle is greater than 900 . If the right
angle and the obtuse angle are added together, the sum is already greater than 1800 , leaving
nothing for the measure of the third angle.
B
3. Make a circle with center at point A and passing through point B.
Draw a diameter through point A and connect point B with the ends
A
of the diameter.
4. The figure
can be divided into two triangles
. Therefore the sum of the
measures of its interior angles is 2 (1800 ) = 3600 .
5. If n = the number of sides in the polygon, the sum of its angles is
1800 ( n − 2 ) = 18000
n − 2 = 10
n = 12
18000
= 1500
12
6. Conjecture: The measure of the sum of the interior angles of an n-sides polygon is
The polygon has 12 sides. Since the polygon is regular, each interior angle is
1800 ( n − 2 ) . This is true since the n-sided polygon can be divided into n − 2 triangles,
each of which is 1800 .
7. Conjecture: The measure of each interior angle of a regular n-sided polygon with n > 3 is
1800 ( n − 2 )
. The sum of all the interior angles of the polygon must be divided by the
n
number of equal angles, n.
8. The number of diagonals that can be drawn from a single vertex is n − 3 . There are n
vertices so it would at first thought seem that there would be n ( n − 3) diagonals altogether.
However, that would be counting each diagonal twice. For example, the diagonal from
vertex A to vertex B would be counted as a vertex from A and as a vertex from B, but there
is really only one vertex between A and B.
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
NOTES
End of Properties of Polygons
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Glossary of Terms
Acute angle – an angle whose measure is between 0 o and 90 o . (Modules: 9, 10)
Acute triangle – a triangle with three acute angles. (Module 10)
Addition Operation – term + term = sum. (Modules: 5 – 10)
Additive Inverse (or opposite of a number, x) – the unique number -x, which when added to x
yields zero. x + (− x) = 0 . (Modules: 4, 8)
Adjacent angles – two angles with the same vertex and a common side between them. Angles
1 and 2 are adjacent angles. (Modules: 9, 10)
1
2
Algebraic Expression – a mathematical combination of constants and variables connected by
arithmetic operations such as addition, subtraction, multiplication, and division.
(Module 8)
Algorithm – a rule (or step by step process) used to help solve a specific type of problem.
(Modules: 5 – 10)
Alternate exterior angles – when a line intersects two parallel lines, eight angles are formed;
two angles that are outside (exterior) the parallel lines and on opposite sides (alternate)
of the intersecting line are called alternate exterior angles. (Module 9)
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MATEMÁTICA August 2006
Alternate interior angles – when a line intersects two parallel lines, eight angles are formed;
two angles that are between (interior) the parallel lines and on opposite sides (alternate)
of the intersecting line are called alternate interior angles. (Module 9)
Altitude – the perpendicular from a vertex to the opposite side (extended if necessary) of a
altitude
altitude
geometric figure. (Module 10)
Angle – the union of two rays with a common endpoint; angles are measured in a counter-
clockwise direction; the angle’s rays are labeled as initial and terminal sides with the
terminal side counter-clockwise from the initial side. (Modules: 9, 10)
terminal side
initial side
Apothem – the apothem of a regular polygon is the radius of an inscribed circle. (Module 10)
apothem
Arc – any part of a circle that can be drawn without lifting the pencil. (Module 10)
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National PASS Center
Area – the measurement in square units of a bounded region. (Module 3)
Associative Property of Addition – this property of real numbers may be written using
variables in the following way: (a + b) + c = a + (b + c) . Terms to be combined may be
grouped in any manner. (Module 4)
Associative Property of Multiplication – this property of real numbers may be written in the
following way: a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c . Terms to be multiplied may be grouped in any
manner. (Module 4)
Axiom – a statement that is accepted as true, without proof. (Module 3)
Base – the numbers being used as a factor in an exponential expression. In the exponential
expression 2 5 , 2 is the base. (Module 7)
Base angles of an isosceles triangle – the angles opposite the equal sides of an isosceles
triangle are the base angles, which are also equal. (Module 10)
Base of an isosceles triangle – the congruent sides of an isosceles triangle are called the legs,
while the third side of the isosceles triangle is called the base. (Module 10)
Binary operation – an operation such as addition, subtraction, multiplication, or division that
changes two values into a single value. (Modules: 5 – 10)
Bisector – a line that divides a figure into two equal parts. (Module 10)
1
Centi – a prefix for a unit of measurement that denotes one one-hundredth ( 100
) of the unit.
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MATEMÁTICA August 2006
Central angle – an angle whose vertex is the center of a circle and whose sides are radii of the
circle. (Module 10)
Chord – a line segment with endpoints on a circle. (Module 10)
Circle – the set of all points in a plane at a given distance (the radius) from a given point (the
center). (Module 10)
radius
center
Circumference – the distance around the edge of a circle. (Modules: 9, 10)
Closed dot – means the number is part of the solution set, thus it is shaded. (Module 8)
Coefficient – the numerical part of a term. (Module 8)
Combine like terms – means to group together terms that are the same (numbers with numbers
/ variables with variables) and are on the same side of the equal sign. (Module 8)
Complementary angles – two angles whose sum is 90 o . (Modules: 9, 10)
Common factor – identical part of each term in an algebraic expression; in the expression ab +
ac, the variable a is the common factor. (Module 8)
Commutative Property of Addition – terms to be combined may be arranged in any order; this
property of real numbers may be written using variables in the following way:
a + b = b + a . (Module 4)
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National PASS Center
Commutative Property of Multiplication – terms to be multiplied may be arranged in any
order; this property of real numbers may be written using variables in the following
way: a ⋅ b = b ⋅ a . (Module 4)
Comparison Axiom – if the first of three quantities is greater than the second and the second is
greater than the third, then the first is greater than the third; if a > b and b > c,
then a > c. (Module 3)
Composite number – a natural number greater than one that has at least one positive factor
other than 1 and itself. (Module 2)
Consecutive even integers – even integers that follow one another such as 2, 4, 6, etc.
(Module 8)
Consecutive integers – integers that follow each other on the number line such as 7, 8, 9, etc.
(Module 8)
Consecutive odd integers – odd integers that follow one another such as 5, 7, 9, etc.
(Module 8)
Constant – any symbol that has a fixed value such as 2 or π. (Modules: 3, 7, 8)
Coplanar – coplanar points are points in the same plane. (Module 9)
Corresponding angles – if a line intersects two parallel lines, eight angles are formed; two
non-adjacent angles that are on the same side of the intersecting line but one between
the parallel lines and one outside the parallel lines are called corresponding angles.
(Module 9)
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MATEMÁTICA August 2006
Counting numbers (or natural numbers) – the set of numbers {1, 2, 3, 4, 5, …}. (Module 1)
Decagon – a ten-sided polygon. (Module 10)
Denominator – the bottom part of a fraction. (Modules: 5, 6, 7, 8)
Diagonal – a line segment with endpoints on two non-consecutive vertices of a polygon.
(Module 10)
Diameter – a line segment that passes through the center of a circle and whose endpoints are
points on the circle. (Module 10)
Difference – the answer to a subtraction problem. (Modules: 5, 6)
Distributive Property of Multiplication over Addition – a property of real numbers used to
write equivalent expressions in the following way: a(b + c) = a ⋅ b + a ⋅ c .
(Modules: 4, 8)
Distributive Property of Multiplication over Subtraction – a property of real numbers used to
write equivalent expressions in the following way: a (b − c) = a ⋅ b − a ⋅ c .
(Modules: 4, 8)
c
a
Dividend – the number being divided in a quotient; in b a or
= c , a is the dividend.
b
(Modules: 5, 6, 7, 8)
Quotient
Division operation – Dividend = Quotient or Divisor Dividend . (Modules: 5, 6, 7, 8)
Divisor
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Elements (of a set) – the objects that belong to a set. (Module 2)
Empty set – a set that has no elements in it. (Module 2)
Equal Quantities Axiom – quantities which are equal to the same quantity or to equal
quantities, are equal to each other. (Module 3)
Equation – a mathematical statement that two quantities are equal to one another. (Module 8)
Equiangular polygon – a polygon with all angles equal. (Module 10)
Equiangular triangle – a triangle with all angles equal. (Module 10)
Equilateral polygon – a polygon with all sides equal. (Module 10)
Equilateral triangle – a triangle with all sides equal. (Module 10)
Existence Property – a property that guarantees a solution to a problem. (Module 4)
Existential quantifier – ∀ is the existential quantifier; it is read “for all,” “for every,” or “for
each.” (Modules: 1, 2)
Exponent – tells how many times a number called the base is used as a factor; in 23 = 8 , three
(3) is the exponent. (Module 7)
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MATEMÁTICA August 2006
Exterior angle – is an angle formed by one side of a polygon and an adjacent side extended.
(Modules: 9, 10)
B
C
D
A
E
Factor – one of the numbers multiplied together in a product; if a ⋅ b = c , then a and b are
factors of c. (Modules: 5, 6)
Fundamental Theorem of Arithmetic – every composite number may be written uniquely
(disregarding order) as a product of primes. (Module 2)
Geometry – the branch of mathematics that investigates relations, properties, and
measurements of solids, surfaces, lines, and angles. (Modules: 9, 10)
Gram (g) – a basic unit of mass in the metric system; 1 gram ≈ .035 ounces.
Heptagon – a seven-sided polygon. (Module 10)
Hexagon – a six-sided polygon. (Module 10)
Hypotenuse – the side opposite the right angle in a right triangle. (Module 10)
Identity – an equation that is true for all values of the variable; every real number is a root of
an identity. (Module 4)
Identity Element for Addition – zero is the additive identity element because 0 may be added
to any number and the number keeps its identity; a + 0 = 0 + a = a for any real number
a. (Module 4)
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Identity Element for Multiplication – one (1) is the multiplicative identity element because
any number may be multiplied by 1 and the number keeps its identity; 1 ⋅ a = a ⋅1 = a for
any real number a. (Module 4)
Improper fraction – a fraction in which the numerator (top #) is larger than the denominator
(bottom #). Improper fractions are greater than 1 and can be turned into mixed
numbers. (Module 5)
Inequality – a mathematical sentence that compares two unequal expressions.
(Modules: 2, 3, 8)
Inscribed angle – an angle whose vertex lies on a circle and whose sides are chords of the
circle. (Module 10)
Integers – the natural numbers, zero, and the additive inverses of the natural numbers;
{…-3, -2, -1, 0, 1, 2, 3…}. (Modules: 1 – 10)
Interior angle – an angle that lies inside a polygon and is formed by two adjacent sides of the
polygon. (Module 10)
Intersect – to cross; two lines in the same plane intersect if and only if they have exactly one
point in common. (Module 9)
Irrational number – a real number that cannot be written as the quotient of two integers; an
irrational number, written as a decimal, does not terminate and does not repeat.
(Module 1)
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MATEMÁTICA August 2006
Isosceles trapezoid – a trapezoid whose non-parallel sides (or legs) are congruent.
(Module 10)
leg
leg
Isosceles triangle – a triangle with two sides equal. (Module 10)
Kilo – a prefix for measurement that denotes one thousand (1000) units.
Kite – a quadrilateral with two pairs of adjacent sides congruent and no opposite sides
congruent. (Module 10)
Least Common Multiple (LCM) – the least common multiple of two or more positive values is
the smallest positive value that is a multiple of each. (Modules: 5, 6)
Legs of an isosceles triangle – the congruent sides of an isosceles triangle are called its legs.
(Module 10)
Like terms – terms which have identical variable factors. (Module 8)
Line – one of the undefined terms; consists of a set of points extending without end in opposite
directions. (Modules: 9, 10)
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Line segment – a subset of a line that contains two points of the line and all points between
those two points. (Modules: 9, 10)
Liter (L) – a basic unit of volume in the metric system; 1 liter ≈ 1.06 liquid quarts.
Lowest common denominator (lcd) (of two or more fractions) – the least common multiple of
the denominators of the fractions. (Modules: 5, 6)
Major arc – an arc of a circle that is greater than a semicircle. (Module 10)
Meter (m) – a basic unit of length in the metric system; 1 meter ≈ 39.37 inches.
1
Milli – a prefix for a unit of measurement that denotes one one-thousandth ( 1000
) of the unit.
Minor arc – an arc of a circle that is less than a semicircle. (Module 10)
Minuend – the number from which something is subtracted; in 5 − 3 = 2 , five (5) is the
minuend. (Modules: 5 – 8)
Multiplicative inverse (or reciprocal of a real number x) – the unique number,
when multiplied by x, yields 1. x ⋅
1
, which,
x
1
= 1 if x ≠ 0 . (Modules: 4, 8)
x
Multiplication operation – factor x factor = product. (Modules: 5 – 8)
Multiplicative property of zero – for any real number a , a ⋅ 0 = 0 ⋅ a = 0 . (Modules: 4 – 8)
Natural numbers (or counting numbers) – the set of numbers {1, 2, 3, 4, 5, …}. (Module 1)
Module 10 – Properties of Polygons
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MATEMÁTICA August 2006
Negative integers – the opposite of the natural numbers. (Modules: 1 – 8)
Nonagon – a nine-sided polygon. (Module 10)
Numerator – the top part of a fraction. (Module 5)
Obtuse angle – an angle that measures between 90 o and 180 o . (Modules: 9, 10)
Obtuse triangle – a triangle with one obtuse angle. (Module 10)
Octagon – an eight-sided polygon. (Module 10)
Open dot – means the number is not part of the solution set, thus it is not shaded. (Module 8)
Parallel lines – lines in the same plane that do not intersect; the two lines are everywhere
equidistant. (Modules: 9, 10)
Parallelogram – a quadrilateral whose opposite sides are parallel. (Module 10)
Pentagon – a five-sided polygon. (Module 10)
Percent – Percent means per 100 or divided by 100. The symbol for percent is %.
(Module 6)
Perfect square – a number whose square root is a natural number. (Module 1)
Perimeter – the sum of the lengths of the sides of a figure or the distance around the figure.
(Modules: 8, 10)
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Perpendicular lines – two lines that form a right angle. (Modules: 9, 10)
Plane – one of the undefined terms; a set of points that form a flat surface extending without
end in all directions. (Modules: 9, 10)
Plane geometry – the branch of mathematics that deals with figures that lie in a plane or flat
surface. (Module 10)
Point – one of the undefined terms; a location with no width, length, or depth.
(Modules: 9, 10)
Polygon – a closed figure bounded by line segments. (Module 9)
Positive integers – the collection of numbers known as natural numbers. (Modules: 1 – 10)
Prime numbers – the natural numbers greater than one (1) that have exactly two factors, one
(1) and themselves. (Module 2)
Product – the result when two or more numbers are multiplied. (Modules: 3 – 10)
Quadrilateral – a polygon with four sides. (Module 10)
Quotient – the number resulting from the division of one number by another. (Modules: 1, 5)
Radical – the symbol that tells you a root is to be taken; denoted by
. (Module 1)
Radicand – the number inside the radical sign whose root is being found; in
radicand. (Module 1)
7x , 7x is the
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Radius (radii) – a line segment with endpoints on the center of the circle and a point on the
circle. (Module 10)
Ratio – proportional relation between two quantities or objects in terms of a common unit.
(Module 5)
Rational numbers – the collection of numbers that can be expressed as the quotient of two
integers; when written as a decimal it will terminate or repeat. (Modules: 1, 5)
Ray – a subset of a line that consists of a point and all points on the line to one side of the
point. (Modules: 9, 10)
Real numbers – the combined collection of the rational numbers and the irrational numbers.
(Module 1)
Reciprocal (or multiplicative inverse of a real number x) – the unique number which, when
1
multiplied by x, yields 1; x ⋅ = 1 if x ≠ 0 . (Module 4)
x
Rectangle – a parallelogram with one right angle. (Modules: 3, 8, 10)
Reflex angle – an angle greater than a straight angle and less than two straight angles.
(Module 9)
Regular polygon – a polygon whose sides and angles are all equal. (Module 10)
Relatively prime – a pair of numbers with no common factor other than 1. (Module 5)
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Repeating decimal – a decimal with an infinite number of digits to the right of the decimal
point created by a repeating set pattern of digits. (Modules: 1, 6)
Rhombus (rhombi) – a parallelogram having two adjacent sides equal. (Module 10)
Right angle – an angle whose sides are perpendicular; having a measure of 90 degrees.
(Modules: 9, 10)
Right triangle – a triangle with one right angle. (Module 10)
Scalene triangle – a triangle with no two sides of equal measure. (Module 10)
Secant – a straight line intersecting a circle in exactly two points. (Module 10)
Sector of a circle – the figure bounded by two radii and an included arc of the circle.
(Module 10)
Sector
Semicircle – an arc equal to half of a circle is called a semicircle. (Module 10)
Set – a collection of objects. (Module 2)
Sides of a polygon – the line segments forming a polygon are called the sides of the polygon.
(Module 10)
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Similar figures – figures with the same shape but not necessarily the same size. (Module 10)
Similar polygons – polygons whose corresponding angles are congruent and whose
corresponding sides are proportional; the symbol ~ is used to indicate that figures are
similar. (Module 10)
Solution – a value that makes the two sides of an equation equal. (Modules: 5 – 10)
Solution set – the set of all roots of the equation. (Module 8)
Square – a rectangle having two adjacent sides equal. (Modules: 8, 10)
Square root – one of the two equal factors of a number. (Module 1)
Straight angle – an angle measuring 180 o . (Modules: 9, 10)
Subset – B is a subset of A, written B ⊆ A, if and only if every element of B is an element of A.
(Module 2)
Substitution Axiom – a quantity may be substituted for its equal in any expression.
(Modules: 3, 4, 7 – 10)
Minuend
Subtraction operation –
− Subtrahend
or
Minuend – Subtrahend = Difference.
Difference
(Modules: 5 – 10)
Subtrahend – the number being subtracted in a subtraction problem; in 5 – 2 = 3, 2 is the
subtrahend. (Modules: 5, 6)
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Sum – the result when two numbers are added. (Modules: 5 – 10)
Supplementary angles – two angles whose sum is 180 o . (Modules: 9, 10)
Term – a single number, a single variable, or a product of a number and one or more variables.
(Modules: 1 – 10)
Terminating decimal – a decimal with a finite (or countable) number of digits to the right of
the decimal point. (Module 6)
Transversal – a straight line that intersects two or more straight lines. (Module 9)
tra nsversa l
Trapezoid – a quadrilateral with exactly one pair of parallel sides. (Module 10)
Triangle – a polygon with three sides. (Modules: 8, 10)
Trichotomy Property – for all real numbers, a and b, exactly one of the following is true;
a = b , a < b , or a > b . (Module 3)
Uniqueness Property – a property that guarantees that when two people work the same
problem they should get the same result. (Module 4)
Universal quantifier – ∃ is the universal quantifier. It is read, there exists or for some.
(Modules: 1, 2)
Variable – a letter or symbol used to represent a number or a group of numbers.
(Modules: 3, 7, 8)
Module 10 – Properties of Polygons
47
MATEMÁTICA August 2006
Vertex – the turning point of a parabola; the common endpoint of the two intersecting rays of
an angle. (Module 10)
Vertex angle of an isosceles triangle – the angle formed by the equal sides of the triangle.
(Module 10)
Vertex of a polygon – a point where two sides of a polygon meet. (Module 10)
Vertical angles – two non-adjacent angles formed by two straight intersecting lines.
(Module 9)
Whole numbers – the collection of natural numbers including zero; {0, 1, 2, 3…}.
(Modules: 1 – 10)
FORMULAS AND DISCOVERIES
The Triangle Inequality:
The sum of two sides of a triangle must be greater than the third side.
In ∆ABC
AB + BC > AC
AB + AC > BC
AC + BC > AB
Integrated Math Concepts
48
National PASS Center
Name
Sketch
Perimeter
Area/
Surface Area
B
c
Triangle
a
P = a+b+c
h
A
C
b
s
D
Square
C
s
s
A
Rectangle
C
l
A= s
P = 2l +2w
A = lw
A
C = 2π r
A =πr
P = 2a + 2b
A = bh
Does not
have
volume
B
l
r
Circle
Parallelogram
h
a
A
D
b
D
b1
s1
2
C
B
P = s1 + b1 + s2 + b2
C
A
Does not
have
volume
Does not
have
volume
( b1 + b2 ) h
Does not
have
volume
A = 12 ap
where p is the
perimeter
Does not
have
volume
S. A. = area of
bases ( B1 + B2 )
+ area of all
lateral faces
V = Bh
or
V = 12 aph
A=
1
2
s2
h
Does not
have
volume
Does not
have
volume
2
w
w
Trapezoid
P = 4s
B
s
D
A = 12 bh
Volume
B
b2
r
s
Regular
Polygon
a
v
t
u
B2
lateral face
Prism
P=r+s+t +
u+v+"
The distance
around a base
B1
Module 10 – Properties of Polygons
49
MATEMÁTICA August 2006
Pyramid
lateralface
h
The distance
around the base
base
Cylinder
S.A. = area of V = 1 Bh
3
the base + area
or
of all the lateral
V = 16 aph
faces.
C = 2π r
S.A. =
2π r 2 + 2π rh
V = π r 2h
l
C = 2π r
S.A. =
π r 2 + 2π rl
V = 13 π r 2 h
r
C = 2π r
S.A. = 4π r 2
V = 43 π r 3
h
r
Cone
h
r
Sphere
End of Glossary
Integrated Math Concepts
50