CliffsNotes Geometry Practice Pack with CD (CliffsNotes (Paperback)) - Softcover

Herzog, David A

Softcover

ISBN 10: 0470488697 ISBN 13: 9780470488690

Verlag: Cliffs Notes, 2010

Inhaltsangabe

About the Contents:

Pretest

Helps you pinpoint where you need the most help and directs you to the corresponding sections of the book

Topic Area Reviews

• Basic geometry ideas

• Parallel lines

• Triangles

• Polygons

• Perimeter and area

• Similar figures

• Right angles

• Circles

• Solid geometry

• Coordinate geometry

Customized Full-Length Exam

Covers all subject areas

Appendix

Postulates and theorems

Über diesen Titel

Über die Autorin bzw. den Autor

David Alan Herzog is the author of numerous books and software programs concerned with test preparation in mathematics and science.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

CliffsNotes Geometry Practice Pack

By David Alan Herzog

John Wiley & Sons

Copyright © 2010 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-48869-0

Chapter One

Basic Geometric Ideas

The word geometry comes from two ancient Greek words, ge, meaning earth, and metria, meaning measure. So, literally, geometry means to measure the earth. It was the first branch of math that began with certain assumptions and used them to draw more complicated conclusions. Over time, geometry has become a body of knowledge that helps us to logically create chains of conclusions that let us go from knowing certain things about a figure to predicting other things about it with certainty. Although a little arithmetic and a little algebra are used in building an understanding of geometry, this branch of math really can stand on its own, as a way of constructing techniques and insights that may help you to better understand later mathematical ideas, and that, believe it or not, may help you to live a more fulfilling life.

Naming Basic Forms

The bulk of this book deals with plane geometry-that is, geometry on a perfectly flat surface. Many different types of plane figures exist, but all of them are made up of a few basic parts. The most elementary of those parts are points, lines, and planes.

Points

A point is the simplest and yet most important building block in geometry. It is a location and occupies no space. Because a point has no height, length, or width, we can't actually draw one. This is true of many geometric parts. We can, however, represent a point, and we use a dot to do that. We name points with single uppercase letters.

This diagram shows three dots that represent points C, M, and Q.

Lines

Lines are infinite series of points. Infinite means without end. A line extends infinitely in two opposite directions, but has no width and no height. Just to be clear, in geometry, line and straight line mean the same thing. Contrary to the popular notion, a line is not the shortest distance between two points. (We'll come back to this later.)

A line may be named by any two points on it, as is line EF, represented by the symbol [??] or [??]. It may also be named by a single lowercase letter, as is line l.

Points that are on the same line are said to be collinear points. Point E and Point F in the preceding diagram are collinear points. Point G is not collinear with E and F. Taken altogether, it may be said that E, F, and G are noncollinear points. You'll see why this distinction is important a little later in this chapter.

Planes

A plane is an infinite set of points extending in all directions along a perfectly flat surface. It is infinitely long and infinitely wide. A plane has a thickness (or height) of zero.

A plane is named by a single uppercase letter and is often represented as a four-sided figure, as in planes U and V in the preceding diagram.

Example Problems

These problems show the answers and solutions.

1. What is the maximum number of lines in a plane that can contain two of the points A, B, and ITLITL?

Answer: 3 Consider that two points name a line. It is possible to make three sets of two points from the three letters: AB, AC, and BC. That means it's possible to form three unique lines: [??], [??], and [??]. See the following figure.

2. What kind of geometric form is the one named H?

Answer: not enough information A single uppercase H could be used to designate a point or a plane.

Postulates and Theorems

As noted at the very beginning of the chapter, geometry begins with assumptions about certain things that are very difficult, if not impossible, to prove and flows on to things that can be proven. The assumptions that geometry's logic is based upon are called postulates. Sometimes, you may see them referred to as axioms. The two words mean essentially the same thing, Here are the first six of them, numbered so that we can refer back to them easily:

Postulate 1: A line contains at least two points. Postulate 2: A plane contains a minimum of three noncollinear points. Postulate 3: Through any two points there can be exactly one line. Postulate 4: Through any three noncollinear points there can be exactly one plane. Postulate 5: If two points lie in a plane, then the line they lie on is in the same plane. Postulate 6: Where two planes intersect, their intersection is a line.

From these six postulates it is possible to prove these theorems, numbered for the same reason:

Theorem 1: If two lines intersect, they intersect in exactly one point. Theorem 2: If a point lies outside a line, then exactly one plane contains the line and the point. Theorem 3: If two lines intersect, then exactly one plane contains both lines.

Example Problems

These problems show the answers and solutions. State the postulate or theorem that may be used to support the statement made about each diagram.

1. There is another point on line l in addition to R.

Answer: A line contains at least two points. (Postulate 1)

2. Only one line contains point M and point N.

Answer: Through any two points there can be exactly one line. (Postulate 3)

Work Problems

Use these problems to give yourself additional practice. State the postulate or theorem that may be used to support the statement made about each diagram.

1. Lines m and l are in the same plane.

2. There is no other intersection for n and p other than B.

3. Point F and [??] are in the same plane.

4. Points J, K, and L are all in the same plane.

5. The intersection of planes P and Q is line r.

6. [??] lies in plane W.

Worked Solutions

1. The figure shows two intersecting lines, and the statement mentions a plane. That relationship is dealt with by Theorem 3: If two lines intersect, then exactly one plane contains both lines.

2. The figure shows two intersecting lines, and the statement mentions the point of intersection. That's covered in Theorem 1: If two lines intersect, they intersect in exactly one point.

3. This figure concerns a line and a noncollinear point, and the statement mentions a plane. That's Theorem 2: If a point lies outside a line, then exactly one plane contains the line and the point. 4. We are shown three noncollinear points, and a plane is mentioned. That's Postulate 4: Through any three noncollinear points there can be exactly one plane. 5. Here, we have two intersecting planes and line r. That's Postulate 6: Where two planes intersect, their intersection is a line. 6. The diagram shows a line in a plane, but two points on that line are clearly marked. That should lead us straight to Postulate 5: If two points lie in a plane, then the line they lie on is in the same plane.

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Details zur Ausgabe

• Verlag: Cliffs Notes
• Erscheinungsdatum: 2010
• Sprache: Englisch
• ISBN 10: 0470488697
• ISBN 13: 9780470488690
• Einband: Taschenbuch
• Auflage: 2
• Anzahl der Seiten: 264
• Kontakt zum Hersteller: HOUGHTON MIFFLIN; https://www.hmhco.com/shop; ; 125 HIGH STREET; BOSTON; MA; 2110; USA
• Verantwortliche Person: Nicht verfügbar