Integrated Math Concepts Module 10. Properties of Polygons. Second Edition. Integrated Math Concepts. Solve Problems. Organize. Analyze. Model.

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Integrated Math Concepts

Module 10

Properties of Polygons

Second Edition

National PASS Center

Integrated Math

Concepts

Solve Problems Organize Model Compute Communicate Measure Reason Analyze

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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.

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Integrated Math Concepts

Module 10

Properties of Polygons

Second Edition

National PASS Center

2006

BOCES Geneseo Migrant Center

27 Lackawanna Avenue

Integrated Math

Concepts

Solve Problems Organize Model Compute Communicate Measure Reason Analyze

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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.

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Module 10 – Properties of Polygons

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.

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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

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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

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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

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

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“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.

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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.

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Properties of

Polygons

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. Integrated Math Concepts Solve Problems Organize Model Compute Communicate Measure Reason Analyze

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An obtuse triangle is a triangle with an obtuse angle. Triangles can be classified according to the measure of their angles.

Think Back

Acute means less than

90°.

Thus all 3 angles are less than 90°.

Think Back

A right triangle contains one angle = 90°.

Think Back

Obtuse means greater than 90°.

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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.

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.

Í

An equiangular triangle is a triangle all of whose angles are equal.

Í

A scalene triangle is a triangle with no two sides congruent.

Í

An equilateral triangle is a triangle all of whose sides are congruent.

Í

An isosceles triangle is a triangle with two sides congruent.

Think Back

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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.

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You should have discovered that the sum of the angles of a triangle is 180 . 0

Example 1

Find the measure of all of the angles in this triangle.

Solution

Since the sum of the angles of a triangle is 180 , then 0

The angles are 35 , 45 , and 100 . 0 0 0

4. How many degrees is each angle of an equiangular triangle? Justify your answer.

5. If a right triangle also has an angle of 60 , how large is the third angle? Justify 0 your answer.

6. Find the measure of all the angles in the following triangle.

An isosceles trapezoid is a trapezoid with one pair of adjacent angles congruent.

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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.

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Side Vertex

polygon

exterior

Interior

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The line segments forming a polygon are called the

sides of the polygon. A point where two sides of a polygon meet is called a vertex of the polygon. A polygon divides a plane into three regions: the interior of the polygon, the exterior of the polygon, and the polygon itself.

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.

Polygon Name Number of Sides or

Vertices triangle 3 quadrilateral 4 pentagon 5 hexagon 6 heptagon 7 octagon 8 nonagon 9 decagon 10 n-gon n

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A diagonal is a line segment joining two non-adjacent vertices of a polygon.

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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

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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.

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

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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 180 , the sum of the measures of the angles in the 0 pentagon is 3 180

(

0

)

=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?

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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

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Í

An exterior angle of a polygon is an angle formed by one side of the polygon and an adjacent side extended.

Example 6

Find the measure of ∠ and the measure of the x exterior angle ∠CDE if the measure of the other angles are as follows:

0 0 0

A x 9 , B 2x 14 , and C 3x 27

∠ = + ∠ = + ∠ = −

Solution

Since the sum of the measure of the interior angles of the trapezoid is 360 , 0

Since ∠ADE is a straight angle,

0 0 0

CDE 180 52 128

∠ = − =

8. Find the measure of the exterior angle y if

0 A 108 ∠ = and ∠ =F 1160. A B C E D

(

0

) (

0

) (

0

)

0 0 0 0 0 3 27 2 14 9 360 7 4 360 7 364 52 x x x x x x x + − + + + + = − = = = y A x - 8 x x x +3 F x - 19 x A B C E D

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

0

37 and 56 . 0

4. Find the measures of all the angles in this triangle if A∠ is half of a right angle, ∠ =B 5x+3, and ∠ =C 6x

5. If the measure of one angle of a triangle is 3 times another and the third angle is 20 larger 0 than the measure of the smaller of the other two angles, what is the measure of all the angles of the triangle?

A B

C

Practice Problems

Properties of Polygons

Directions: Write your answers in your math journal. Label this exercise Properties of Polygons.

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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

11. Find the values of all the angles in this sketch if AEF is a straight angle and

x A B C D E 3 2 D x 8, C 2x 5, B x 10, and A x 13. ∠ = + ∠ = + ∠ = + ∠ = +

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12. If the figure is a regular hexagon, find the measure of x∠ . (Hint: What do the angle markings mean?)

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?

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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 360 . Justify your answer. 0

5. The sum of the angles in a regular polygon is 1800 . How many sides does the polygon 0 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

(

3

)

2 n n−

diagonals. Justify this answer. (Hint: How many diagonals are there from each vertex? Why? How many vertices does an n-sided figure have? Why is it necessary to divide by 2?)

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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 equiangular triangle. Since the other angles have the same measure, each angle is 60 . 0

5. A right angle has a measure of 90 . Let x = the 0 measure of the third angle. The angle is 30 . 0

6. The sum of the angles is 180 : 0

0 0 0 180 3 180 60 x x x x x + + = = = 0 0 0 0 0 0 90 60 180 150 180 30 x x x + + = + = = 0 0 0 0 0 0 0 0 0 2 40 55 180 3 15 180 3 165 55 2 40 70 x x x x x x − + + = + = = = − =

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7.

8.

But the question asks for the measure of y.

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

(

)

0

3 180

108

5 = . Since an exterior angle is supplementary

0− 0 = 0 scalene triangle square isosceles trapezoid rectangle rhombus equilateral triangle trapezoid parallelogram isosceles triangle triangle polygon quadrilateral

(

)

(

)

0 0 0 0 0 0 180 19 180 140 19 59 y x y y = − − = − − =

(

)

(

)

(

) (

)

0 0 0 0 0 0 0 0 0 0 108 8 3 116 19 5 180 5 200 900 5 700 140 x x x x x x x x + − + + + + + + − = + = = =

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Explorations

#1. You should have discovered that the sum of all three angles of a triangle is 180°. #2.

0

8 180 =1440 . 4.

=10800

(

)

Number of sides on polygon Number of triangles

6 4 7 5 8 6 9 7 10 8 11 9 n n−2

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Answers to Practice Problems

Connections and Modeling

1. The polygons are equilateral triangles, rhombi, and squares.

2. Classified by the measure of their angles Classified by the measure of their sides Equiangular triangles Scalene triangles Acute triangles Equilateral triangles Right triangles Isosceles triangles Obtuse triangles

3. Let x = the measure of the third angle. The third angle is 87 . 0

4. 1

( )

0 0 2 A 90 45 ∠ = =

(

)

0 0 0 0 0 0 0 0 0 0 45 5 13 6 180 11 48 180 11 132 12 5 3 63 6 72 x x x x x x x + + + = + = = = + = =

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.

The angles are 32 , 0 96 , and 0 52 . 0

6. a. True b. False c. False d. False e. True f. False g. True h. 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).

0 0 0 0 0 0 37 56 180 93 180 87 x x x + + = + = = 0 0 0 A 45 , B 63 , and C 72 ∠ = ∠ = ∠ = 0 0 0 0 0 0 0 3 20 180 5 160 32 3 96 20 52 x x x x x x x + + + = = = = + =

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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

(

)

0

5 180

128.6

7 ≈ .

An exterior angle is approximately 1800−128.60 =51.40.

b. An interior angle of a regular octagon is

(

)

0 0 6 180 135 8 = . An exterior angle is 1800−1350 =450.

c. An interior angle of a regular nonagon is

(

)

d. An interior angle of a regular decagon is

(

)

11. Since ∠ and AEDx ∠ are supplementary, ∠AED=1800− . x

(

) (

)

(

)

0 3 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 13 10 2 5 8 180 3 180 2 26 3 20 4 10 2 16 360 2 2 540 9 432 1080 9 648 72 x x x x x x x x x x x x x + + + + + + + + − = + + + + + + + + − = + = = = 0 72 x ∠ = 0 0 0 0 A x 13 72 13 85 ∠ = + = + =

( )

0 0 0 0 3 3 2 2 B x 10 72 10 118 ∠ = + = + =

( )

0 0 0 0 C 2x 5 2 72 5 149 ∠ = + = + = 0 0 0 0 D x 8 72 8 80 ∠ = + = + = 0 0 0 0 AED 180 x 180 72 108 ∠ = − = − =

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0 0 0 0 2 120 180 2 60 30 y y y + = = =

12. Each interior angle in a regular hexagon is

(

)

0

4 180

120

6 = .

The marked angles in the triangle are equal. If y = the measure of one of these angles, then

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 360 . 0

b. Each figure is divided into two triangles. The sum of the measures of the angles of a quadrilateral is 360 . 0

2. a. The angles have the same measure. b. The angles have the same measure.

c. The polygons will begin to look more like a circle.

4. a.

b.

5 diagonals 9 diagonals 20 diagonals c. Conjecture: An n-gon has

(

3

)

2 n n−

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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 90 . If the right 0

angle and the obtuse angle are added together, the sum is already greater than 180 , leaving 0 nothing for the measure of the third angle.

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 of the diameter.

4. The figure can be divided into two triangles . Therefore the sum of the measures of its interior angles is 2 180

(

0

)

=3600.

5. If n = the number of sides in the polygon, the sum of its angles is

The polygon has 12 sides. Since the polygon is regular, each interior angle is

0

1800

150 12 = 6. Conjecture: The measure of the sum of the interior angles of an n-sides polygon is

(

)

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Glossary of Terms

Acute angle – an angle whose measure is between 0o and 90o. (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+ − = . (Modules: 4, 8) ( x) 0

Adjacent angles – two angles with the same vertex and a common side between them. Angles

1 and 2 are adjacent angles. (Modules: 9, 10)

2 1

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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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

geometric figure. (Module 10)

al

ti

tud

e

altitude

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)

initial side

terminal side

Apothem – the apothem of a regular polygon is the radius of an inscribed circle. (Module 10)

apothem

(45)

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+ + = + + . Terms to be combined may be ) c a (b c) 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⋅ ⋅ = ⋅ ⋅ . Terms to be multiplied may be grouped in any ( ) (a b c) 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 25_{, 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)

(46)

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 90o. (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:

(47)

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)

(48)

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)

Dividend – the number being divided in a quotient; in

c b a or a c b = , a is the dividend. (Modules: 5, 6, 7, 8) Division operation – Quotient Dividend

Quotient or Divisor Dividend

(49)

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= , three 8

(50)

Exterior angle – is an angle formed by one side of a polygon and an adjacent side extended. (Modules: 9, 10) A B C E D

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

(51)

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)

(52)

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

(53)

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.

Milli – a prefix for a unit of measurement that denotes one one-thousandth 1 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, 1

x, which, when multiplied by x, yields 1. x 1 1

x

⋅ = if x≠0. (Modules: 4, 8)

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)

(54)

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 90o and 180o. (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.

(55)

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)

(56)

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

multiplied by x, yields 1; x 1 1 x

⋅ = if x≠0. (Module 4)

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)

(57)

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)

(58)

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 180o. (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)

Subtraction operation –

Minuend Subtrahend Difference

− or Minuend – Subtrahend = Difference.

(Modules: 5 – 10)

Subtrahend – the number being subtracted in a subtraction problem; in 5 – 2 = 3, 2 is the

(59)

Sum – the result when two numbers are added. (Modules: 5 – 10)

Supplementary angles – two angles whose sum is 180o. (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 nsve rsa 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.

(60)

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 + > + > + >

(61)

Name Sketch Perimeter Area/

Surface Area Volume

Triangle P a b c= + + 1 2 A= bh Does not have volume Square P = 4s A = 2 s Does not have volume Rectangle P = 2l +2w A = lw Does not have volume Circle C 2 r= π _A 2 r π

= Does not have volume P 2= a+2b A bh= Does not have volume 1 1 2 2 P s= + + +b s b 1

(

)

1 2 2 A= b +b h Does not have volume P = r + s + t + u + v + " A = 1 2ap where p is the perimeter Does not have volume The distance around a base S. A. = area of bases (B₁+B₂) + area of all lateral faces V Bh= or 1 2 V= aph s s s s D C B A A B C D l l w w A B C a b c h r A B C D h 1 b 2 b 1 s s2 1 B 2 B lateral face Parallelogram Trapezoid Regular Polygon Prism a r s t u v A B C D h b a

(62)

The distance around the base

S.A. = area of the base + area of all the lateral faces. 1 3 V= Bh or 1 6 V= aph C 2 r= π S.A. = 2 2πr +2πrh 2 V=πr h C 2 r= π S.A. = 2 ₂ r rl π + π V=13πr h2 Sphere C 2 r= π S.A. = 4 rπ 2 V= 43πr3 End of Glossary h base lateralface h r r h l r Pyramid Cylinder Cone

Integrated Math Concepts Module 10. Properties of Polygons. Second Edition. Integrated Math Concepts. Solve Problems. Organize. Analyze. Model.

Integrated Math Concepts

Module 10

Properties of Polygons

Second Edition

National PASS Center

Integrated Math

Concepts

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

Integrated Math Concepts

Module 10

Properties of Polygons

Second Edition

National PASS Center

BOCES Geneseo Migrant Center

27 Lackawanna Avenue

Integrated Math

Concepts

Module 10 – Properties of Polygons

Table of Contents

Integrated Math Concepts – Introduction

The PASS Concept

Strategy

Course Organization

Testing

Test taking tips

Properties of

Polygons

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Review

Practice Problems

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