Here are some conjectures based on the given patterns using inductive reasoning: a. The number of dots in each successive figure increases by one in both the row and column. Therefore, the next figure will have 4 dots arranged in a 2 by 2 square. b. The dot inside the diamond moves counterclockwise to the next vertex, while the dot inside the rectangle moves clockwise to the next vertex. Therefore, the next figure will show the dot in the diamond at the bottom right vertex and the dot in the rectangle at the top left vertex. c. Each term is obtained by doubling the previous term. Therefore, the next term should be 32. d. Each term is obtained by multiplying the previous

1. NEXT
GENERATION III
MATH
(Textbook)

2. Next Generation Math III
Textbook
Philippine Copyright 2011 by DIWA LEARNING SYSTEMS INC
All rights reserved. Printed in the Philippines
Editorial, design, and layout by University Press of First Asia
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ISBN 978-971-46-0184-0
Authors
Maria 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-Reviewer
Lorelei 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.

3. 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.

4. Table of Contents
Unit I The Essentials of Geometry
Chapter 1 The Art of Reasoning
Lesson 1 The Reasons Behind ...................................................................................... 2
Lesson 2 If-and-Then Statements................................................................................. 8
Lesson 3 The Converse, the Inverse, the Contrapositive,
and the Biconditional Statements ................................................................ 13
Lesson 4 Deductive Reasoning ................................................................................... 18
IT Matters ................................................................................................................... 25
Chapter 2 The Models of Points, Lines, and Planes
Lesson 1 Points, Lines, and Planes in Space .............................................................. 26
Lesson 2 Pairs of Lines .............................................................................................. 34
Lesson 3 Basic Postulates on Points, Lines, and Planes.............................................. 40
Lesson 4 Segment Relationships ................................................................................ 45
IT Matters ................................................................................................................... 52
Chapter 3 All about Angles
Lesson 1 Angles ......................................................................................................... 57
Lesson 2 Angle Measurement ..................................................................................... 63
Lesson 3 Bisector of an Angle .................................................................................... 70
Lesson 4 Pairs of Angles ............................................................................................ 76
IT Matters ................................................................................................................... 83
Chapter 4 The Transversal
Lesson 1 Angles Formed by a Transversal Line........................................................... 87
Lesson 2 Perpendicular Lines..................................................................................... 94
Lesson 3 Parallel Lines ............................................................................................ 100
IT Matters ................................................................................................................. 105
Chapter 5 The Polygon
Lesson 1 Polygons.................................................................................................... 109
Lesson 2 Angles in a Polygon ................................................................................... 115
IT Matters ................................................................................................................. 121

5. Unit II The Triangles
Chapter 6 The Triangle Congruence
Lesson 1 Properties of a Triangle .............................................................................. 126
Lesson 2 Congruence on Triangles ........................................................................... 133
Lesson 3 Congruent Triangles .................................................................................. 138
IT Matters ................................................................................................................. 148
Chapter 7 Inequalities in Triangles
Lesson 1 Inequalities in a Triangle ........................................................................... 153
Lesson 2 Inequalities in Two Triangles ..................................................................... 158
IT Matters ................................................................................................................. 164
Unit III Quadrilaterals and Similarity
Chapter 8 Quadrilaterals
Lesson 1 Properties of Parallelograms ..................................................................... 166
Lesson 2 More on Parallelograms ............................................................................ 172
Lesson 3 Properties of Special Quadrilaterals .......................................................... 178
Lesson 4 The Trapezoid and Its Properties ............................................................... 185
Lesson 5 The Kite and Its Properties ........................................................................ 189
Lesson 6 Solving Problems Involving Quadrilaterals ................................................. 193
IT Matters ................................................................................................................. 197
Chapter 9 Similarity
Lesson 1 Proportional Segments .............................................................................. 198
Lesson 2 The Basic Proportionality Theorem and Its Converse ................................. 205
Lesson 3 Other Proportionality Theorems ................................................................. 211
IT Matters ................................................................................................................. 219
Chapter 10 More on Similarity
Lesson 1 Similar Polygons........................................................................................ 221
Lesson 2 Similar Triangles ....................................................................................... 227
Lesson 3 Triangle Similarity Theorems ..................................................................... 234
Lesson 4 Similarities in a Right Triangle .................................................................. 241
Lesson 5 The Pythagorean Theorem and Its Converse .............................................. 248
Lesson 6 Special Right Triangles .............................................................................. 254
Lesson 7 Areas and Perimeters of Similar Triangles.................................................. 258
Lesson 8 Solving Word Problems on Similarity ......................................................... 262
IT Matters ................................................................................................................. 268

6. Unit IV Circles, Plane and Solid Geometry, and Plane Coordinate
Geometry
Chapter 11 Circles
Lesson 1 Parts of a Circle ......................................................................................... 271
Lesson 2 Arcs and Central Angles ............................................................................ 276
Lesson 3 Inscribed Angles ........................................................................................ 281
Lesson 4 Properties of a Line Tangent to a Circle...................................................... 287
Lesson 5 Theorems on Chords of a Circle ................................................................. 293
Lesson 6 Angles Formed by Tangents and Secants ................................................... 298
Lesson 7 Tangent Circles and Common Tangents..................................................... 305
IT Matters ................................................................................................................. 312
Chapter 12 Plane and Solid Geometry
Lesson 1 Circumference of a Circle, Perimeter and Area of Plane Figures ................ 313
Lesson 2 Surface Area and Volume of Solid Figures ................................................. 319
IT Matters ................................................................................................................. 326
Chapter 13 Plane Coordinate Geometry
Lesson 1 The Cartesian Coordinate Plane: A Review ................................................. 328
Lesson 2 Equation of a Line and the Point of Intersection of Two Lines..................... 333
Lesson 3 Parallel and Perpendicular Lines in the Coordinate Plane .......................... 337
Lesson 4 Distance and Midpoint Formulas............................................................... 340
Lesson 6 Circles in the Coordinate Plane.................................................................. 346
IT Matters ................................................................................................................. 350
Glossary ................................................................................................................. 351
Bibliography ................................................................................................................. 358
Index ................................................................................................................. 360

7. The Essentials of Unit I
Geometry
This unit covers five 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 figures 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.

8. Chapter 1
THE ART OF REASONING
Learning Objectives
• Define and differentiate the kinds of reasoning
• Define 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
Lesson 1
The Reasons Behind
Power Up
1. Look closely at each figure. Which is longer, P or X?
X
P
2 Next Generation Math III

9. 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, definitions, 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 figure is • .
• • • • • •
• • • • • •
• • • •
b. A line has points.
A triangle has lines.
A triangle has points. EXTEND THE CONCEPT
Solution: Scientific Method
a. Inductive reasoning The scientific method uses inductive reasoning
b. Deductive reasoning and deductive reasoning. It consists of defining 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.
The Essentials of Geometry 3

10. 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
4 Next Generation Math III

11. 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 first 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 first four terms, then 15 is the fifth 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.
The Essentials of Geometry 5

12. _____________ 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
6 Next Generation Math III

13. 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.)
The Essentials of Geometry 7