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NEXT GENERATION

MATH III

(Textbook)

Next Generation Math IIITextbook

Philippine Copyright 2011 by DIWA LEARNING SYSTEMS INCAll rights reserved. Printed in the Philippines

Editorial, design, and layout by University Press of First Asia

No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopying, recording, or any information storage and retrieval systems, without permission in writing from the copyright owner.

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ISBN 978-971-46-0184-0

AuthorsMaria Maitas M. Marasigan finished her master’s degree in Mathematics and bachelor’s degree in Mathematics for Teachers from De La Salle University–Dasmariñas and Philippine Normal University (PNU), respectively. She is a licensed college instructor and has been teaching various Mathematics subjects in several universities and colleges both in Manila and in Cavite for more than 12 years. She is presently taking up her doctorate degree in Mathematics Education at PNU and is a full-time academic faculty of Lyceum University of the Philippines–Cavite since June 2008.

Angelo D. Uy obtained his bachelor’s degree in Mathematics for Teachers at Philippine Normal University, where he is also currently pursuing his master’s degree in Education with specialization in Mathematics Education. He was a trainer in different Math competitions and a participant in various seminar-workshops sponsored by the Mathematics Teachers Association of the Philippines. He taught at the grade school and high school departments of Hotchkiss Learning Center in Surigao del Sur and at the grade school department of De La Salle Santiago Zobel School in Ayala Alabang. Mr. Uy presently teaches Mathematics at Jacobo Z. Gonzales Memorial National High School (JZGMNHS) in Biñan, Laguna. He is currently the JZGMNHS Math Club adviser and one of the PROs of the Secondary Mathematics Teachers Association of Laguna (SEMATAL).

Consultant-ReviewerLorelei B. Ladao-Saren obtained her master’s degree in Mathematics, with high distinction, from De La Salle University (DLSU)–Dasmariñas and her bachelor’s degree in Statistics from University of the Philippines–Diliman. She is presently pursuing her doctorate degree in Mathematics Education at Philippine Normal University. Ms. Ladao-Saren was a former director for Research, Publication, and Community Extension Services at World Citi Colleges. She has also taught Mathematics at Asia Pacific College, Southville Foreign University, and at DLSU–Dasmariñas. She currently teaches Mathematics at DLSU–College of St. Benilde and at the Graduate School of Rizal Technological University.

Preface

The Next Generation Math series covers topics and competencies that are aligned with the Basic Education Curriculum (BEC) and the Engineering and Science Education Program (ESEP) of the Department of Education. It is composed of different mathematics disciplines: elementray algebra in first year; intermediate algebra in second year; geometry in third year; and advanced algebra, trigonometry, statistics, and calculus in fourth year. It tries to cover the many important topics that will satisfy the needs of different groups of learners.

The series supports the constructivist approach to teaching and learning process. Lessons are presented through meaningful activities which are designed to provide you an opportunity to make different connections between a concrete situations and mathematics. The activities are designed to develop your skills in problem solving, critical thinking, decision making, and creative thinking through exchange of ideas and own discovery. Each book in this series provides opportunities for you to discuss, explore, and construct mathematical ideas and interpret new information and knowledge at a different perspective. You will also be able to structure and evaluate your own conjectures and apply previously acquired knowledge and skills.

The series has the following salient features:• Lessons are inquiry based, enriched with applicable technologies, and integrated with

science and real-life applications. • Emphasis on the development of higher-order thinking skills is evident on the illustrative

examples and exercises provided in every lesson. To enhance your mathematics skills, the degree of difficulty of the problems ranges from simple to more challenging ones.

• Exercises include research work to emphasize the importance of research as a tool in satisfying quest for knowledge and acquiring valuable insights about certain topics.

• Historical notes, application of mathematical ideas in future careers, and pieces of trivia are presented in each chapter.It is with a sincere desire to provide a useful tool in enhancing appreciation and better

understanding of mathematics that the Next Generation Math series was conceptualized.

Table of Contents

Unit I The Essentials of Geometry

Chapter 1 The Art of ReasoningLesson 1 The Reasons Behind ...................................................................................... 2Lesson 2 If-and-Then Statements ................................................................................. 8Lesson 3 The Converse, the Inverse, the Contrapositive, and the Biconditional Statements ................................................................ 13Lesson 4 Deductive Reasoning ................................................................................... 18IT Matters ................................................................................................................... 25

Chapter 2 The Models of Points, Lines, and Planes Lesson 1 Points, Lines, and Planes in Space .............................................................. 26Lesson 2 Pairs of Lines .............................................................................................. 34Lesson 3 Basic Postulates on Points, Lines, and Planes .............................................. 40Lesson 4 Segment Relationships ................................................................................ 45IT Matters ................................................................................................................... 52

Chapter 3 All about AnglesLesson 1 Angles ......................................................................................................... 57Lesson 2 Angle Measurement ..................................................................................... 63Lesson 3 Bisector of an Angle .................................................................................... 70Lesson 4 Pairs of Angles ............................................................................................ 76IT Matters ................................................................................................................... 83

Chapter 4 The TransversalLesson 1 Angles Formed by a Transversal Line ........................................................... 87Lesson 2 Perpendicular Lines ..................................................................................... 94Lesson 3 Parallel Lines ............................................................................................ 100IT Matters ................................................................................................................. 105

Chapter 5 The PolygonLesson 1 Polygons .................................................................................................... 109Lesson 2 Angles in a Polygon ................................................................................... 115IT Matters ................................................................................................................. 121

Unit II The Triangles

Chapter 6 The Triangle CongruenceLesson 1 Properties of a Triangle .............................................................................. 126Lesson 2 Congruence on Triangles ........................................................................... 133Lesson 3 Congruent Triangles .................................................................................. 138IT Matters ................................................................................................................. 148

Chapter 7 Inequalities in TrianglesLesson 1 Inequalities in a Triangle ........................................................................... 153Lesson 2 Inequalities in Two Triangles ..................................................................... 158IT Matters ................................................................................................................. 164

Unit III Quadrilaterals and Similarity

Chapter 8 QuadrilateralsLesson 1 Properties of Parallelograms ..................................................................... 166Lesson 2 More on Parallelograms ............................................................................ 172Lesson 3 Properties of Special Quadrilaterals .......................................................... 178Lesson 4 The Trapezoid and Its Properties ............................................................... 185Lesson 5 The Kite and Its Properties ........................................................................ 189Lesson 6 Solving Problems Involving Quadrilaterals ................................................. 193IT Matters ................................................................................................................. 197

Chapter 9 SimilarityLesson 1 Proportional Segments .............................................................................. 198Lesson 2 The Basic Proportionality Theorem and Its Converse ................................. 205Lesson 3 Other Proportionality Theorems ................................................................. 211IT Matters ................................................................................................................. 219

Chapter 10 More on SimilarityLesson 1 Similar Polygons ........................................................................................ 221Lesson 2 Similar Triangles ....................................................................................... 227Lesson 3 Triangle Similarity Theorems ..................................................................... 234Lesson 4 Similarities in a Right Triangle .................................................................. 241Lesson 5 The Pythagorean Theorem and Its Converse .............................................. 248Lesson 6 Special Right Triangles .............................................................................. 254Lesson 7 Areas and Perimeters of Similar Triangles .................................................. 258Lesson 8 Solving Word Problems on Similarity ......................................................... 262IT Matters ................................................................................................................. 268

Unit IV Circles, Plane and Solid Geometry, and Plane Coordinate Geometry

Chapter 11 CirclesLesson 1 Parts of a Circle ......................................................................................... 271Lesson 2 Arcs and Central Angles ............................................................................ 276Lesson 3 Inscribed Angles ........................................................................................ 281Lesson 4 Properties of a Line Tangent to a Circle ...................................................... 287Lesson 5 Theorems on Chords of a Circle ................................................................. 293Lesson 6 Angles Formed by Tangents and Secants ................................................... 298Lesson 7 Tangent Circles and Common Tangents ..................................................... 305IT Matters ................................................................................................................. 312

Chapter 12 Plane and Solid GeometryLesson 1 Circumference of a Circle, Perimeter and Area of Plane Figures ................ 313Lesson 2 Surface Area and Volume of Solid Figures ................................................. 319IT Matters ................................................................................................................. 326

Chapter 13 Plane Coordinate GeometryLesson 1 The Cartesian Coordinate Plane: A Review ................................................. 328Lesson 2 Equation of a Line and the Point of Intersection of Two Lines ..................... 333Lesson 3 Parallel and Perpendicular Lines in the Coordinate Plane .......................... 337Lesson 4 Distance and Midpoint Formulas ............................................................... 340Lesson 6 Circles in the Coordinate Plane .................................................................. 346IT Matters ................................................................................................................. 350

Glossary ................................................................................................................. 351Bibliography ................................................................................................................. 358Index ................................................................................................................. 360

This unit covers fi ve chapters that deal with the art of reasoning, points, lines, planes, angles, transversal lines, and polygons.

In Chapter 1, you will learn new ways to reason mathematically and to use the different kinds of reasoning in formulating conjectures. Chapter 2 will help you understand the different models of points, lines, and planes in space, and how they can be used to determine the relationships between segments and to prove theorems. In Chapter 3, you will learn the different kinds of angles based on their measurements and the different geometric fi gures that can be modeled by the different pairs of angles. Chapter 4 will introduce the angles formed by a transversal line, and the perpendicularity and parallelism of lines. Finally, Chapter 5 will cover convex and concave polygons and the angles in a polygon. All of these topics are essential tools in understanding geometry.

Unit I The Essentials of Geometry

2 Next Generation Math III

Chapter 1

THE ART OF REASONING

Lesson 1The Reasons Behind

1. Look closely at each fi gure. Which is longer, P or X?

Learning Objectives• Defi ne and differentiate the kinds of reasoning• Defi ne and apply the different methods of proving• Apply the symbols of logic in solving problems• Use analogies in descriptions, comparisons, and understanding new concepts and

ideas• Write a conjecture based on inductive and deductive reasonings• Understand a conditional statement written in an alternate form and rewrite it in the “if-then form”• Write and determine the truth value of the converse, inverse, and contrapositive of a

given conditional statement• Determine whether a biconditional statement is true or false• Prove logical arguments deductively• Solve real-life problems where there is possibility of more than one adequate solution• Give reasons for the choices made as well as vary the style and amount of detail in

explanations• Demonstrate an advanced ability to read and write different conditional statements

into the “if-then” form• Solve problems with an optimistic attitude and open mind

Power Up

P

X

The Essentials of Geometry 3

2. What conjecture can you make out of this statement? Point is to segment as arrow is to _______.

3. What would be the next set of numbers?

1

1 2 2 1 1 3 4 3 1 1 4 7 7 4 1

How did you get your answer?

4. Consider the following statements:

A student likes geometry. Maria is a student.

Are the two statements related? If so, how? What made you say so? What can you conclude from the two statements?

Walk Through

• Statement – expression with a complete thought• Hypothesis – statement where a conclusion is being drawn• Conclusion – relation that can be drawn from the given statements• Conjecture – unproven statement or proposition that is based on observations • Reasoning – process of determining the relation between the given hypotheses

Kinds of Reasoning• Inductive reasoning – reasoning that is based on observing and recognizing patterns in a

set of data and using the patterns to arrive at a conjecture• Deductive reasoning – reasoning that uses facts, rules, defi nitions, or properties in a

logical order to show that a desired conclusion is true

Example 1: Determine the kind of reasoning used in each statement below.

a. Given: • • • ? Thus, the next fi gure is • . • • • • • • • • • • • • • • • • b. A line has points. A triangle has lines. A triangle has points.

Solution: a. Inductive reasoning b. Deductive reasoning

Scientifi c MethodThe scientifi c method uses inductive reasoning

and deductive reasoning. It consists of defi ning the problem, making an educated guess or hypothesis, gathering facts from observations and experiments, classifying data, logically drawing conclusions, and proving the hypothesis to reach conclusions.

EXTEND THE CONCEPT


Example 2: Identify the hypothesis and the conclusion of each statement.

a. Given: Thus,

b. Given: x – 2 = 6. Thus, x = 8. c. A plane is a closed figure. A square is a plane. A square is a closed figure. d. Given the first five terms of the sequence 3, 5, 7, 9, 11. Then, the sixth term is 13.

Solution:

a. Hypothesis: Conclusion:

b. Hypothesis: x – 2 = 6 Conclusion: x = 8 c. Hypotheses: A plane is a closed figure. A square is a plane. Conclusion: A square is a closed figure. d. Hypothesis: 3, 5, 7, 9, 11 Conclusion: 13 is the sixth term

Example 3: Supply a word or a number that will make each statement true.

a. Circumference is to circle as _____ is to square. b. _____ is to angle as meter is to length. c. 2, 4, 5, 10, 11, __, 23 d. 9:3 = __:5 e. If x – 5 = 6, then x = ___.

Solution: a. perimeter b. Degree or Radian c. 22 d. 15 e. 11

Example 4: Make a conjecture based on inductive reasoning. Justify your conclusion.

a.

b.

c. 1, 2, 4, 8, 16 d. 1, 2, 6, 24, 120


Solution:

a. The number of dots in each row and column is one more than the number of dots in the previous row and column.

b. The dot inside the diamond rotates counterclockwise, while the dot inside the rectangle rotates clockwise.

c. 32. Each term after the fi rst is obtained by doubling the previous term.

d. 720. Each term is obtained by applying the following solution: 1 × 2 = 2, 2 × 3 = 6, 6 × 4 = 24, 24 × 5 = 120, 120 × 6 = 720

Example 5: Write a conclusion using deductive reasoning. a. All students who passed algebra are enrolled in geometry class. Carlo passed algebra. b. The number of vertices in a polygon also determines the number of sides. A square has 4 vertices. c. Lines that meet at a common point are intersecting lines. Line m and line n meet at a common point. d. An acute angle measures less than 90. ∠K is an acute angle.

Note: In this book, we shall consider angle measures in degrees.

Solution: a. Carlo is enrolled in geometry class. b. A square has 4 sides. c. Line m and line n are intersecting lines. d. ∠K measures less than 90.

Move Up

I. Identify whether the conclusion was a result of induction or deduction._____________ 1. A quadrilateral has four sides. A rectangle has four sides. A rectangle is a quadrilateral._____________ 2. If an angle is obtuse, then it cannot be acute. ∠M is obtuse. ∠M cannot be acute._____________ 3. If 3, 6, 9, and 12 are the fi rst four terms, then 15 is the fi fth term._____________ 4. Ray is to an angle as side is to a triangle._____________ 5. Trees can be found in a forest. Birds live on trees. Therefore, birds can be found in a forest._____________ 6. If 3:6, 6:18, and 12:54, then 24:162.


_____________ 7. All animals are mortal. All humans are animals. Therefore, all humans are mortal.

_____________ 8. If a polygon has five sides, then it is a pentagon.

9. All trees are plants. _____________ All acacia are trees. Thus, all acacia are plants.

_____________ 10. If a number is divisible by 2, then it is even.

II. Encircle the letter that corresponds to the correct answer.

1. Which of the following is not a process? a. statement b. analogy c. deduction d. induction

2. Which reasoning is based on facts and not on patterns? a. deduction c. analogy b. induction d. none of these

3. Which of the following figures have the same length?

i. ii. iii. iv.

a. i and ii b. ii and iv c. i and iii d. ii and iii

For numbers 4 and 5, consider the statement:

If two points determine a line, then three distinct points determine a plane.

4. Which of the following is the hypothesis? a. Three points determine a line. c. Three points determine a plane. b. Three distinct points determine d. Two points determine a line. a plane.

5. Which of the following is the conclusion? a. Three points determine a line. c. Two points determine a plane. b. Three distinct points determine d. Two points determine a line. a plane.

For numbers 6 and 7, pick another pair that has the same relationship as the given pair.

6. 3:9 as _________. a. 9:3 b. 3:1 c. 12:36 d. 12:60

7. Thermometer is to temperature as _________ is to _________. a. telescope, astronomy c. clock, minutes b. scale, weight d. microscope, biologist


8. Determine the next figure in the given set of figures below.

a. b. c. d.

For numbers 9 and 10, consider the two statements below.

The number 11 is a prime number. A prime number has factors of 1 and itself.

9. What conclusion can you draw out of the given statements? a. The number 11 is a prime number. b. The number 11 is factorable. c. The number 11 has factors of 1 and itself. d. The number 11 is a factor.

10. What kind of reasoning should be used to draw the conclusion? a. deduction c. analogy b. induction d. none of these

III. Read and analyze each problem carefully.

1. What can you conclude about the size range of carrots if you measure 20 carrots and found that they are all between 6 and 8 inches long? Which type of reasoning are you going to apply? Justify your answer.

2. As you enter a room, you suddenly observe that the room is dark. You wonder why the room is dark and you attempt to find explanations out of curiosity. You thought of a lot of possibilities why the room is dark. You might think that the lights are turned off or the room’s light bulb has burnt out, and worst, you could be going blind. What are you going to do to test your hypotheses? Will you apply inductive or deductive reasoning? Justify your answer.

3. Look for an article that shows how deductive and inductive reasonings are used in scientific studies. Share it with the class and let the class formulate conjectures from the article. Cite your references. (For example: If 300 out of 1 000 people got better in taking vitamins, then 30% of any population might also get better in taking vitamins.)