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


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1 Solve Problems Analyze Organize Reason Integrated Math Concepts Model Measure Compute Communicate Integrated Math Concepts Module 1 Properties of Polygons Second Edition National PASS Center 26
2 National PASS Center BOCES Geneseo Migrant Center 27 Lackawanna Avenue Mount Morris, NY 1451 (585) (585) (fax) 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 2 Education Service Center, San Antonio, Texas. Copyright 26 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.
3 Solve Problems Analyze Organize Reason Integrated Math Concepts Model Measure Compute Communicate Integrated Math Concepts Module 1 Properties of Polygons Second Edition National PASS Center 26 BOCES Geneseo Migrant Center 27 Lackawanna Avenue Mount Morris NY 1451
4 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 (24) and Yesenia Medina of San Juan, Texas, and Wild Rose, Wisconsin (26), 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 26. 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
5 Module 1 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
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7 Integrated Math Concepts Introduction The PASS Concept PASS (Portable Assisted Study Sequence) is a study program created to help you earn credit through semiindependent 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
8 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 welldefined 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
9 Module 5 Fractions Become comfortable with fractions by understanding their makeup 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: Onestep equations Twostep equations Complex equations (combining like terms, use of the distributive property, variables on both sides) Multistep equations Translate algebraic inequalities Solve and graph solutions to one and twostep inequalities iii
10 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 1 Properties of Polygons Recognize and classify 2dimensional shapes circles, triangles and quadrilaterals Find 2dimensional 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 A set is a collection of objects. 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
11 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
12 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
13 National PASS Center Solve Problems Reason Analyze Organize Integrated Math Concepts Model Properties of Polygons Measure Compute Communicate Objectives: Recognize and classify 2dimensional shapes circles, triangles and quadrilaterals Find 2dimensional 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 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 twodimensional figures. You are no doubt already familiar with many of these figures: circles, squares, rectangles, and triangles, etc. 1 Module 1 Properties of Polygons
14 MATEMÁTICA August 26 A circle is the set of all points in a plane at a given distance (the radius) from a given point (the center). center radius 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 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 2
15 National PASS Center A triangle is a polygon having three sides. Triangles can be classified according to the measure of their angles. An acute triangle is a triangle with three acute angles. Think Back Acute means less than 9. Thus all 3 angles are less than 9. A right triangle is a triangle with a right angle. Think Back A right triangle contains one angle = 9. An obtuse triangle is a triangle with an obtuse angle. Think Back Obtuse means greater than 9. Thus only one angle needs to be greater than 9. 3 Module 1 Properties of Polygons
16 MATEMÁTICA August 26 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. A scalene triangle is a triangle with no two sides congruent. An isosceles triangle is a triangle with two sides congruent. 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 4
17 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 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. 5 Module 1 Properties of Polygons
18 MATEMÁTICA August 26 You should have discovered that the sum of the angles of a triangle is 18. Example 1 Find the measure of all of the angles in this triangle. 7x Solution Since the sum of the angles of a triangle is ( ) The angles are 18, then 12x x+ 7x= 18 28x + 4 = 18 28x = 14 x = 5 7x = 35 9x = 45 12x + 4 = 1 35, 45, and 1. 12x + 4 9x 4. How many degrees is each angle of an equiangular triangle? Justify your answer. 5. If a right triangle also has an angle of your answer. 6, how large is the third angle? Justify C 2x 4 6. Find the measure of all the angles in the following triangle. A x 55 B Integrated Math Concepts 6
19 National PASS Center A quadrilateral is a polygon with four sides. Quadrilaterals can be divided into subcategories by using their characteristics. A parallelogram is a quadrilateral whose opposite sides are parallel A rhombus is a parallelogram with all sides congruent. A rectangle is a parallelogram with all angles congruent. A square is a parallelogram with the same characteristics as the rhombus and rectangle, where all sides and all angles are congruent. A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a trapezoid with one pair of adjacent angles congruent. 7 Module 1 Properties of Polygons
20 MATEMÁTICA August 26 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 8
21 National PASS Center The 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 polygon divides a plane into three regions: the interior of the polygon, the exterior of the polygon, and the polygon itself. Interior polygon exterior Side 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 1 ngon n A diagonal is a line segment joining two nonadjacent vertices of a polygon. 9 Module 1 Properties of Polygons
22 MATEMÁTICA August 26 Example 3 How many diagonals does a triangle have? Solution A triangle has no diagonals because there are no nonadjacent 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 1
23 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 n An interior angle of a polygon is an angle that lies inside the polygon and is formed by two adjacent sides of the polygon. 11 Module 1 Properties of Polygons
24 MATEMÁTICA August 26 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 pentagon is ( ) 3 18 = , the sum of the measures of the angles in the 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 12
25 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 Module 1 Properties of Polygons
26 MATEMÁTICA August 26 An exterior angle of a polygon is an angle formed by one side of the polygon and an adjacent side extended. A B C 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: A = x+ 9, B = 2x+ 14, and C = 3x 27 A x D E Solution Since the sum of the measure of the interior angles of the trapezoid is x+ ( 3x 27 ) + ( 2x+ 14 ) + ( x+ 9 ) = 36 Since 7x 4 = 36 7x = 364 x = 52 36, ADE is a straight angle, CDE = = 128 A x  19 y 8. Find the measure of the exterior angle y if A = 18 and F = 116. x  8 x x x +3 F Integrated Math Concepts 14
27 National PASS Center 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 18 = 72. Since a regular hexagon has 6 equal angles, each interior angle is angle is =. Since exterior angles are supplementary to interior angles, each exterior = Find the measure of an interior and an exterior angle in a regular pentagon. 15 Module 1 Properties of Polygons
28 MATEMÁTICA August 26 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 16
29 National PASS Center 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. 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 37 and Practice Problems Properties of Polygons Directions: Write your answers in your math journal. Label this exercise Properties of Polygons Find the measures of all the angles in this triangle if A of a right angle, B= 5x + 3, and C= 6x is half C A B 5. If the measure of one angle of a triangle is 3 times another and the third angle is 2 larger than the measure of the smaller of the other two angles, what is the measure of all the angles of the triangle? 17 Module 1 Properties of Polygons
30 MATEMÁTICA August 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? 1. 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 D = x+ 8, C = 2x+ 5, B = x+ 1, and A = x A E x F Integrated Math Concepts 18
31 National PASS Center 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 nonadjacent 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? 19 Module 1 Properties of Polygons
32 MATEMÁTICA August 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 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 ngon where n is greater than 3. Integrated Math Concepts 2
33 National PASS Center 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 36. Justify your answer. 5. The sum of the angles in a regular polygon is 18. 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 nsided 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 nsided polygon if n > 3. You may have written your conjecture in this ( 3) n n form: An nsided polygon has diagonals. Justify this answer. (Hint: How many 2 diagonals are there from each vertex? Why? How many vertices does an nsided figure have? Why is it necessary to divide by 2?) 21 Module 1 Properties of Polygons
34 MATEMÁTICA August 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 6. x+ x+ x= 18 3x = 18 x = 6 5. A right angle has a measure of measure of the third angle. The angle is 9. Let x = the x = x = 18 x = 3 6. The sum of the angles is 18 : 2x 4 + x+ 55 = 18 3x + 15 = 18 3x = 165 x = 55 2x 4 = 7 Integrated Math Concepts 22
35 National PASS Center 7. polygon triangle quadrilateral scalene triangle isosceles triangle parallelogram trapezoid equilateral triangle rhombus rectangle isosceles trapezoid square 8. ( x ) x x ( x ) ( x ) ( ) = x + 2 = 9 5x = 7 x = 14 ( ) ( ) But the question asks for the measure of y. y = 18 x 19 y = y = 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 ( ) 3 18 of a regular pentagon is 5 to this angle, an exterior angle has a measure of = 18. Since an exterior angle is supplementary = Module 1 Properties of Polygons
36 MATEMÁTICA August 26 Explorations #1. You should have discovered that the sum of all three angles of a triangle is 18. #2. #3. 1. The measure of the sum of the interior angles of a hexagon is ( ) 4 18 = The measure of the sum of the interior angles of an octagon is ( ) 6 18 = The measure of the sum of the interior angles of a decagon is 4. Number of sides on polygon ( ) 8 18 = 144. Number of triangles n n 2 No. of sides in polygon No. of diagonals from a vertex Sum of interior angles 2 18 = ( ) 5 3 ( ) 3 18 = ( ) 4 18 = ( ) 6 18 = ( ) 8 18 = 144 Integrated Math Concepts 24
37 Connections and Modeling Answers to Practice Problems National PASS Center 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. x = 18 The third angle is 87. x + 93 = 18 1 A= 9 = 45 x = ( ) ( x ) x= 18 11x + 48 = 18 11x = 132 x = 12 5x + 3 = 63 6x = 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 A = 45, B = 63, and C = 72 x + 2. x+ 3x+ x+ 2 = 18 5x = 16 x = 32 3x = 96 x + 2 = 52 The angles are 32, 96, and 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). 25 Module 1 Properties of Polygons
38 MATEMÁTICA August 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. ( 5 18 ) 1. a. An interior angle of a regular heptagon is An exterior angle is approximately b. An interior angle of a regular octagon is An exterior angle is = 45. c. An interior angle of a regular nonagon is An exterior angle is = 4. d. An interior angle of a regular decagon is 11. Since x and An exterior angle is = 36. AED are supplementary, = ( ) ( ) ( ) = 135. = 14. = 144. AED = 18 x. ( x+ 13 ) + ( 3 2 x+ 1 ) + ( 2x+ 5 ) + ( x+ 8 ) + ( 18 x) = 3( 18 ) 2x x x x x= 2( 54 ) x = 72 A= x + 13 = = 85 ( ) B = x + 1 = = ( ) C = 2x + 5 = = 149 D= x + 8 = = 8 AED = 18 x = = 18 9x = 18 9x = 648 x = 72 Integrated Math Concepts 26
39 National PASS Center 12. Each interior angle in a regular hexagon is The marked angles in the triangle are equal. If y = the measure of one of these angles, then ( ) = 12. 2y + 12 = 18 2y = 6 y = 3 Furthermore since x y + = 12, x= 12 y = 12 3 = 9 Explorations 1. a. Triangles The sum of the measures of the angles of a rectangle is 36. b. Each figure is divided into two triangles. The sum of the measures of the angles of a quadrilateral is a. The angles have the same measure. b. The angles have the same measure. c. The angles are all 9 (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 2 18 ( ) Sum of the measure of interior angles = ( ) = ( ) = ( ) = 864 Measure of each interior angle = 54 = = = 27 Module 1 Properties of Polygons
40 MATEMÁTICA August = ( ) 7 68 ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = = = = = = b. There is a limit. The values are approaching because an interior angle must be less than a straight angle. c. The polygons will begin to look more like a circle. 18, but they cannot reach that value 4. a. b. 5 diagonals 9 diagonals 2 diagonals c. Conjecture: An ngon has ( 3) n n 2 diagonals. Integrated Math Concepts 28
41 National PASS Center 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 angle and the obtuse angle are added together, the sum is already greater than nothing for the measure of the third angle. 9. If the right 18, leaving 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. A B 4. The figure can be divided into two triangles. Therefore the sum of the measures of its interior angles is ( ) 2 18 = If n = the number of sides in the polygon, the sum of its angles is 18 The polygon has 12 sides. Since the polygon is regular, each interior angle is = 6. Conjecture: The measure of the sum of the interior angles of an nsides polygon is ( ) 18 2 n. This is true since the nsided polygon can be divided into n 2 triangles, each of which is Conjecture: The measure of each interior angle of a regular nsided polygon with n > 3 is ( n ) 18 2 ( n ) 18 2 = 18 n 2= 1 n = 12. 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. 29 Module 1 Properties of Polygons
42 MATEMÁTICA August 26 NOTES End of Properties of Polygons Integrated Math Concepts 3
43 National PASS Center Glossary of Terms Acute angle an angle whose measure is between o and 9 o. (Modules: 9, 1) Acute triangle a triangle with three acute angles. (Module 1) Addition Operation term + term = sum. (Modules: 5 1) Additive Inverse (or opposite of a number, x) the unique number x, which when added to x yields zero. x+ ( x) =. (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, 1) 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 1) 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) 31 Module 1 Properties of Polygons
44 MATEMÁTICA August 26 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 1) altitude altitude Angle the union of two rays with a common endpoint; angles are measured in a counterclockwise direction; the angle s rays are labeled as initial and terminal sides with the terminal side counterclockwise from the initial side. (Modules: 9, 1) terminal side initial side Apothem the apothem of a regular polygon is the radius of an inscribed circle. (Module 1) apothem Arc any part of a circle that can be drawn without lifting the pencil. (Module 1) Integrated Math Concepts 32
45 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 1) 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 1) Binary operation an operation such as addition, subtraction, multiplication, or division that changes two values into a single value. (Modules: 5 1) Bisector a line that divides a figure into two equal parts. (Module 1) 1 Centi a prefix for a unit of measurement that denotes one onehundredth ( 1) of the unit. 33 Module 1 Properties of Polygons
46 MATEMÁTICA August 26 Central angle an angle whose vertex is the center of a circle and whose sides are radii of the circle. (Module 1) Chord a line segment with endpoints on a circle. (Module 1) Circle the set of all points in a plane at a given distance (the radius) from a given point (the center). (Module 1) center radius Circumference the distance around the edge of a circle. (Modules: 9, 1) 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 9 o. (Modules: 9, 1) 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) Integrated Math Concepts 34
47 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 nonadjacent 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) 35 Module 1 Properties of Polygons
48 MATEMÁTICA August 26 Counting numbers (or natural numbers) the set of numbers {1, 2, 3, 4, 5, }. (Module 1) Decagon a tensided polygon. (Module 1) Denominator the bottom part of a fraction. (Modules: 5, 6, 7, 8) Diagonal a line segment with endpoints on two nonconsecutive vertices of a polygon. (Module 1) Diameter a line segment that passes through the center of a circle and whose endpoints are points on the circle. (Module 1) 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: ab ( + c) = ab + 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: ab ( c) = ab a c. (Modules: 4, 8) Dividend the number being divided in a quotient; in (Modules: 5, 6, 7, 8) c ba or a c b =, a is the dividend. Division operation Dividend Divisor Quotient Quotient or Divisor Dividend =. (Modules: 5, 6, 7, 8) Integrated Math Concepts 36
49 National PASS Center 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 1) Equiangular triangle a triangle with all angles equal. (Module 1) Equilateral polygon a polygon with all sides equal. (Module 1) Equilateral triangle a triangle with all sides equal. (Module 1) 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 (3) is the exponent. (Module 7) 3 2 = 8, three 37 Module 1 Properties of Polygons
50 MATEMÁTICA August 26 Exterior angle is an angle formed by one side of a polygon and an adjacent side extended. (Modules: 9, 1) B C A D 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, 1) Gram (g) a basic unit of mass in the metric system; 1 gram.35 ounces. Heptagon a sevensided polygon. (Module 1) Hexagon a sixsided polygon. (Module 1) Hypotenuse the side opposite the right angle in a right triangle. (Module 1) 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 may be added to any number and the number keeps its identity; a+ = + a= a for any real number a. (Module 4) Integrated Math Concepts 38
51 National PASS Center 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 1) Integers the natural numbers, zero, and the additive inverses of the natural numbers; { 3, 2, 1,, 1, 2, 3 }. (Modules: 1 1) Interior angle an angle that lies inside a polygon and is formed by two adjacent sides of the polygon. (Module 1) 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) 39 Module 1 Properties of Polygons
52 MATEMÁTICA August 26 Isosceles trapezoid a trapezoid whose nonparallel sides (or legs) are congruent. (Module 1) leg leg Isosceles triangle a triangle with two sides equal. (Module 1) Kilo a prefix for measurement that denotes one thousand (1) units. Kite a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. (Module 1) 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 1) 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, 1) Integrated Math Concepts 4
53 National PASS Center Line segment a subset of a line that contains two points of the line and all points between those two points. (Modules: 9, 1) Liter (L) a basic unit of volume in the metric system; 1 liter 1.6 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 1) Meter (m) a basic unit of length in the metric system; 1 meter inches. 1 Milli a prefix for a unit of measurement that denotes one onethousandth ( )of the unit. 1 Minor arc an arc of a circle that is less than a semicircle. (Module 1) 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. 1 x = 1 if x. (Modules: 4, 8) x Multiplication operation factor x factor = product. (Modules: 5 8) Multiplicative property of zero for any real number a, a = a =. (Modules: 4 8) Natural numbers (or counting numbers) the set of numbers {1, 2, 3, 4, 5, }. (Module 1) 41 Module 1 Properties of Polygons
54 MATEMÁTICA August 26 Negative integers the opposite of the natural numbers. (Modules: 1 8) Nonagon a ninesided polygon. (Module 1) Numerator the top part of a fraction. (Module 5) Obtuse angle an angle that measures between 9 o and 18 o. (Modules: 9, 1) Obtuse triangle a triangle with one obtuse angle. (Module 1) Octagon an eightsided polygon. (Module 1) 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, 1) Parallelogram a quadrilateral whose opposite sides are parallel. (Module 1) Pentagon a fivesided polygon. (Module 1) Percent Percent means per 1 or divided by 1. 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, 1) Integrated Math Concepts 42
55 National PASS Center Perpendicular lines two lines that form a right angle. (Modules: 9, 1) Plane one of the undefined terms; a set of points that form a flat surface extending without end in all directions. (Modules: 9, 1) Plane geometry the branch of mathematics that deals with figures that lie in a plane or flat surface. (Module 1) Point one of the undefined terms; a location with no width, length, or depth. (Modules: 9, 1) Polygon a closed figure bounded by line segments. (Module 9) Positive integers the collection of numbers known as natural numbers. (Modules: 1 1) 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 1) Quadrilateral a polygon with four sides. (Module 1) 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 Module 1 Properties of Polygons 43
56 MATEMÁTICA August 26 Radius (radii) a line segment with endpoints on the center of the circle and a point on the circle. (Module 1) 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, 1) 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. (Module 4) x Rectangle a parallelogram with one right angle. (Modules: 3, 8, 1) 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 1) Relatively prime a pair of numbers with no common factor other than 1. (Module 5) Integrated Math Concepts 44
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