Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    From $11.99/month after trial. Cancel anytime.

    Painless Geometry
    Painless Geometry
    Painless Geometry
    Ebook447 pages3 hours

    Painless Geometry

    By Barron's Educational Series and Lynette Long

    4.5/5

    ()

    Read preview

    About this ebook

    Learning at home is now the new normal. Need a quick and painless refresher? Barron’s Painless books make learning easier while you balance home and school. 

    Barron's makes learning Geometry fun and PAINLESS!

    Painless Geometry provides lighthearted, step-by-step learning and includes:
    • Characteristics of distinct shapes, such as circles, quadrilaterals, and triangles
    • Discussion on how geometric principles can solve real-world problems
    • Painless tips, common pitfalls, instructive tables, diagrams, “brain tickler” quizzes and answers throughout each chapter, and more.
    LanguageEnglish
    PublisherBarrons Educational Services
    Release dateAug 10, 2020
    ISBN9781506268057
    Painless Geometry

    Read more from Barron's Educational Series

    Related to Painless Geometry

    Related ebooks

    Study Aids & Test Prep For You

    View More
    View More

    Related podcast episodes

    Related articles

    • UNLIMITED

      How I Rewired My Brain to Become Fluent in Math: The building blocks of understanding are memorization and repetition.

      Sep 15, 2016

      I was a wayward kid who grew up on the literary side of life, treating math and science as if they were pustules from the plague. So it’s a little strange how I’ve ended up now—someone who dances daily with triple integrals, Fourier transforms, and t

      11 min read

    • UNLIMITED

      Curves, Ellipses, and Spirals

      Jun 7, 2018

      Curves, Ellipses, and Spirals
      5 min read

    • UNLIMITED

      Drawing Circles

      Feb 8, 2018

      Drawing Circles
      4 min read

    • UNLIMITED

      Loosen Your Lines

      Oct 29, 2021

      Loosen Your Lines
      4 min read

    • UNLIMITED

      Develop Your Skills

      Jun 16, 2023

      Develop Your Skills
      2 min read

    • UNLIMITED

      Learn To Draw What You See

      Jul 10, 2020

      Learn To Draw What You See
      4 min read

    • UNLIMITED

      Have No Fear

      Jul 21, 2023

      Have No Fear
      7 min read

    • UNLIMITED

      3. Foreshortening

      Oct 1, 2021

      Beloved by some and frustrating to others, foreshortening describes the principles of diminution applied to a single subject. Whether you are drawing a log in a landscape or a figure in repose, the part of the subject that is closest to you will appe

      4 min read

    • UNLIMITED

      One Line At A Time

      Jun 11, 2021

      ■ Draw what you see, not what you think you see ■ Measure and compare your subjects ■ Don’t erase your mistakes, draw over them Many people think that drawing is a skill you are born with rather than one you learn. It’s true that a few seem to have a

      5 min read

    • UNLIMITED

      Kid Stuff

      Nov 2, 2023

      Kid Stuff
      2 min read

    • UNLIMITED

      Plans vs Proportions

      Sep 10, 2020

      Plans vs Proportions
      6 min read

    • UNLIMITED

      The Bare Bones

      Jan 22, 2020

      The Bare Bones
      7 min read

    • UNLIMITED

      Do The Maths

      Sep 2, 2024

      by Robyn Arianrhod (UNSW Press, $49.99) Imagine an arrow on a computer screen –in PowerPoint, say. You can drag it around the screen and it’s the same arrow. You can make different arrows by stretching it or rotating it to point in a different direc

      2 min read

    • UNLIMITED

      Deciphering Cable Patterns

      Sep 29, 2022

      Deciphering Cable Patterns
      8 min read

    • UNLIMITED

      Carving Claw Feet

      Mar 7, 2023

      It’s safe to say that the highlight of the Newport tea table is the legs that support it. That’s because at the base of those legs resides wonderfully carved claw feet. So to help you make the tea table dazzle, we’ll walk you through what it takes to

      4 min read

    • UNLIMITED

      4. Linear Perspective

      Oct 29, 2021

      4. Linear Perspective
      4 min read

    • UNLIMITED

      Drawing Faces An Introduction

      Sep 2, 2024

      Drawing Faces An Introduction
      3 min read

    • UNLIMITED

      5 Stages of Line Drawing

      Oct 25, 2019

      When talking about the nature of the actual lines that are used when drawing any freehand composition, there are to my way of thinking five clearly separate stages you go through to take your layout from a blank piece of paper to the precise and accu

      6 min read

    • UNLIMITED

      Monumental Yet Accessible

      Nov 28, 2023

      Deb Bennett, PhD, has been a columnist and consulting editor at EQUUS for more than three decades. During that time, she has helped thousands of readers better understand their horses from the skeleton out—what makes them work and how humans can best

      4 min read

    • UNLIMITED

      Classic Compositional Approaches

      Jul 30, 2020

      Classic Compositional Approaches
      2 min read

    • UNLIMITED

      Adding Curves to Your Toolkit

      Dec 18, 2019

      Adding Curves to Your Toolkit
      8 min read

    • UNLIMITED

      Classic Compositional Approaches

      Jun 23, 2020

      Classic Compositional Approaches
      2 min read

    Related categories

    Reviews for Painless Geometry

    4.5/5

    3 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Painless Geometry - Barron's Educational Series

      Chapter 1

      A Painless Beginning

      Geometry is a mathematical subject, but it is just like a foreign language. In geometry, you have to learn a whole new way of looking at the world. There are hundreds of terms in geometry that you’ve probably never heard before. Geometry also uses many common terms, but they have different meanings. A point is no longer the point of a pencil, and a plane is not something that flies in the sky. To master geometry, you have to master these familiar and not so familiar terms, as well as the theorems and postulates that are the building blocks of geometry. You have a big job in front of you, but if you follow the step-by-step approach in this book and do all the experiments, it can be painless.

      Undefined Terms

      Undefined terms are terms that are so basic they cannot be defined, but they can be described.

      A point is an undefined term. A point is a specific place in space. A point has no dimension. It has no length, width, or depth. A point is represented by placing a dot on a piece of paper. A point is labeled by a single letter.

      A line is a set of points that extend indefinitely in either direction. In this book, the term line will always mean a straight line.

      images/himg-11-1.png

      This line is called line AB, or line AC, or line BC, or AB↔ , AC↔, or BC↔. The two-ended arrow over the letters tells you that the letters are describing a line. You can name a line by any two points that lie on the line.

      A plane is a third undefined term. A plane is a flat surface that extends infinitely in all directions. If you were to imagine a tabletop that went in every direction forever, you would have a plane.

      CAUTION—Major Mistake Territory!

      The term line always refers to a straight line. This is not a line because it is not straight.

      images/himg-12-1.png

      A line also extends forever in both directions. A line has no endpoints. This is not a line because it does not extend forever in both directions.

      images/himg-12-2.png

      A plane is often represented by a four-sided figure. Place a capital letter in one of the corners of the figure to name the plane. This is plane P.

      images/himg-12-3.png

      Some Defined Terms

      A line segment is part of a line with two endpoints. A line segment is labeled by its endpoints. We use two letters with a bar over them to indicate a line segment. This line segment is segment XY or XY¯.

      images/himg-13-1.png

      A ray is a part of a line that has one endpoint and extends infinitely in the other direction. The endpoint and one other point on the ray label a ray. This ray could be labeled ray AB or ray AC. When you label a ray, put a small arrow on the top of the two letters to indicate that it is a ray. The arrow on top of the two letters should show the direction of the ray. This ray could also be labeled as AB→ or AC→. This is not BC→ since B is not an endpoint.

      images/himg-13-2.png

      Opposite rays are two rays that have the same endpoint and form a straight line. Ray BA and ray BC are opposite rays. We can also say that BA→ and BC→ are opposite rays.

      images/himg-13-3.png

      Parallel lines are two lines in the same plane that do not intersect.

      images/himg-13-4.png

      CAUTION—Major Mistake Territory!

      No matter how far you extend two parallel lines, they will not intersect. These two lines do not intersect here, but they are not parallel. If you continue to extend them, they will intersect.

      images/himg-14-1.png

      Collinear points lie on the same line. A, B, and C are collinear points. D is not collinear with A, B, and C.

      images/himg-14-2.png

      Noncollinear points do not lie on the same line. X, Y, and Z are noncollinear points. All three of them cannot lie on the same line.

      images/himg-14-3.png

      BRAIN TICKLERSSet # 1

      Decide whether each of the following statements is true or false.

      1.A ray has one endpoint.

      2.Points that lie on the same line are called linear.

      3.A line is always straight.

      4.A line segment can be ten miles long.

      (Answers are on page 14.)

      More Defined Terms

      An angle is a pair of rays that have the same endpoint.

      images/himg-15-1.png

      The rays are the sides of the angles. The endpoint where the two rays meet is called the vertex of the angle.

      A polygon is a closed figure with three or more sides that intersect only at their endpoints. The sides of the polygons are line segments. The points where the sides of a polygon intersect are called the vertices of the polygons.

      These are all polygons.

      images/himg-15-2.png

      Two shapes are congruent if they have exactly the same size and shape.

      Each of these pairs of figures is congruent.

      images/himg-15-3.png

      CAUTION—Major Mistake Territory!

      These two shapes are not congruent. They have the same shape, but they are not the same size.

      images/himg-16-1.png

      The perimeter of a polygon is the sum of the lengths of all the sides of the polygon. The perimeter of a polygon is expressed in linear units such as inches, feet, meters, miles, and centimeters.

      The area of a geometric figure is the number of square units the figure contains. The area of a figure is written in square units such as square inches, square feet, square miles, square centimeters, square meters, and square kilometers.

      The ratio of two numbers a and b is a divided by b, written ab as long as b is not equal to zero.

      A proportion is an equation that sets two ratios equal. ab=cd is an example of a proportion.

      BRAIN TICKLERSSet # 2

      Determine whether each of the following statements is true or false.

      1.A polygon can be composed of three segments.

      2.The perimeter of a figure can be measured in square inches.

      3.5/0 is a ratio.

      (Answers are on page 14.)

      Postulates

      Postulates are generalizations in geometry that cannot be proven true. They are just accepted as true. Do the following experiment to discover one of the most basic postulates of geometry.

      EXPERIMENT

      Find out how two points determine a line.

      Materials

      Pencil

      Paper

      Ruler

      Procedure

      1.Draw two points on a piece of paper. To draw a point make a dot on the paper as if you were dotting the letter i.

      2.Draw a line through these two points. Now draw another line through these same two points. Draw a third line through these two points. Are the three lines the same or different?

      3.Draw two other points. How many different lines can you draw through these two points?

      Something to think about . . .

      How many lines can you draw through a single point?

      POSTULATE Two points determine a single line.

      What does this postulate mean?

      You can draw many lines through a single point.

      You can only draw one line through any two different points.

      POSTULATE At least three points not on the same line are needed to determine a single plane.

      What does this postulate mean?

      Through a single point there are an infinite number of planes.

      Through two points you can also find an infinite number of planes.

      Through three points that lie in a straight line there are an infinite number of planes.

      Through three points that do not lie in a straight line there is only one plane.

      POSTULATE If two different planes intersect, they intersect on exactly one line.

      What does this postulate mean?

      Not all planes intersect.

      Two different planes cannot intersect on more than one line.

      Theorems

      Theorems are generalizations in geometry that can be proven true.

      THEOREM If two lines intersect, they intersect at exactly one point.

      What does this theorem mean?

      Not all lines intersect.

      Two different lines cannot intersect at more than one point.

      This statement is a theorem and not a postulate because mathematicians can prove this statement. They do not have to accept it as true.

      THEOREM Through a line and a point not on that line, there is only one plane.

      Three noncollinear points determine a single plane. A line provides two points and the point not on that line is the third noncollinear point.

      Conditional Statements

      Conditional statements are statements that have the form "If p, then q."

      An example of a conditional statement is If a figure is a square, then it has four sides.

      The first part of the conditional statement is called the hypothesis. In the above example, If a figure is a square is the hypothesis. The second part of the conditional statement is called the conclusion. In the above example, then it has four sides is the conclusion. Many theorems are conditional statements. If the first part of the statement is true, then the second part of the statement has to be true.

      The converse of a conditional statement is formed by reversing the hypothesis and the conclusion. The converse of the conditional statement If a figure is a square, then it has four sides is If a figure has four sides, then it is a square. The original statement is true, but the converse is not true. A rectangle has four sides, but it is not a square.

      PAINLESS TIP

      Write each postulate or theorem on a separate index card. On the back of the index card, draw a diagram to illustrate the postulate or theorem. Memorize all of them!

      Geometric Proofs

      In geometry, there are two types of proofs—indirect proofs and deductive proofs.

      Indirect proofs

      In an indirect proof, you assume the opposite of what you want to prove true. This assumption leads to an impossible conclusion, so your original assumption must be wrong.

      Example:

      You want to prove three is an odd number.

      Assume the opposite. Assume three is an even number.

      All even numbers are divisible by two.

      Three is not divisible by two, so it can’t be an even number.

      Natural numbers are either even or odd, so three must be odd.

      Deductive proofs

      Deductive proofs are the classic geometric proofs. Deductive reasoning uses definitions, theorems, and postulates to prove a new theorem true. A typical deductive proof uses a two-column format. The statements are on the left and the reasons are on the right.

      In this book, deductive proofs will have a different format. Three questions will be used to construct the proof. These three questions will help you think about proofs in a painless way.

      1.What do you know?

      2.What can you infer based on what you know?

      3.What can you conclude?

      Geometric Symbols

      Much of geometry is written in symbols. You have to understand these symbols if you are going to understand geometry. Here are some common geometric symbols.

      BRAIN TICKLERSSet # 3

      Name the following symbols.

      1.≅

      2.<

      3.⊥

      4.≥

      (Answers are on page 14.)

      PAINLESS TIP

      Write each of the basic geometry symbols on the front of an index card. Write the word or words each symbol represents on the back. Memorize each of the symbols using the cards for help.

      SUPER BRAIN TICKLERS

      Match these definitions or symbols to the correct terms.

      (Answers are on page 14.)

      BRAIN TICKLERS—THE ANSWERS

      Set # 1, page 4

      1.True

      2.False

      3.True

      4.True

      Set # 2, page 6

      1.True

      2.False

      3.False

      Set # 3, page 13

      1.Congruent to

      2.Less than

      3.Perpendicular to

      4.Greater than or equal to

      Super Brain Ticklers, page 13

      1.C

      2.D

      3.E

      4.F

      5.A

      6.H

      7.B

      8.G

      Chapter 2

      Enjoying the preview?
      Page 1 of 1
      pFad - Phonifier reborn

      Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

      Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


      Alternative Proxies:

      Alternative Proxy

      pFad Proxy

      pFad v3 Proxy

      pFad v4 Proxy