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

Herzog, David Alan

Softcover

ISBN 10: 0470488697 ISBN 13: 9780470488690

Publisher: Cliffs Notes Inc.,U.S., 2010

Synopsis

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

About this title

About the Author

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

Excerpt. © Reprinted by permission. All rights reserved.

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.

Finding Segments, Midpoints, and Rays

You'll recall that geometry means earth measure. We've already dealt with the concept of lines, but because lines are infinite, they can't be measured. Much of geometry deals with parts of lines. Some of those parts are very special-so much so that they have their own special names and symbols. The first such part is the line segment.

Line Segments

A line segment is a finite portion of a line and is named for its two endpoints.

In the preceding diagram is segment [bar.ST]. Notice the bar above the segment's name. Technically, [bar.ST] refers to points S and T and all the points in between them. ST, without the bar, refers to the distance from S to T. You'll notice that [bar.ST] is a portion of [??].

Each point on a line or a segment can be paired with a single real number, which is known as that point's coordinate. The distance between two points is the absolute value of the difference of their coordinates.

If b > a, then AB = b - a. This postulate, number 7, is known as the Ruler Postulate.

Example Problems

These problems show the answers and solutions.

1. Find the length of [bar.EH], or, put more simply, find EH.

Answer: 6 To find the length of [bar.EH], first find the coordinates of point E and point H.

E's coordinate is 3, and point H's coordinate is 9.

EH = 9 - 3

EH = 6

2. Find the length of [bar.FK], or, put more simply, find FK.

Answer: 10 To find FK, first find the coordinates of point F and point K.

F's coordinate is 5, and point K's coordinate is 15.

FK = 15 - 5

FK = 10

Segment Addition and Midpoint

Postulate 8 is known as the Segment Addition Postulate. It goes like this:

Postulate 8 (Segment Addition Postulate): If N lies between M and P on a line, then MN + NP = MP. This is, in fact, one of many postulates and theorems that can be restated in general terms as the whole is the sum of its parts.

The midpoint of a line segment is the point that's an equal distance from both endpoints. B is the midpoint of [bar.AC] because [bar.AB] = [bar.BC].

This brings us to the obvious fact stated by Theorem 4:

Theorem 4: A segment has exactly one midpoint.

Example Problems

These problems show the answers and solutions.

1. V lies between Q and S. Find QS if QV = 6 and VS = 10.

Answer: 16 Since V lies between Q and S, Postulate 8 tells us that

QV + VS = QS 6 + 10 = 16 QS = 16

2. Find the midpoint of [bar.RZ].

Answer: V We can solve this in two ways. First consider the coordinates of the endpoints, R (7) and Z (31). Their difference is 24 (31 - 7 = 24). That means that the segment is 24 units long, so its midpoint must be half of 24, or 12 units from either endpoint. Adding 12 to the 7 (R's coordinate) gives us 19, the coordinate of V. V is the midpoint.

The other method is to take the average of the coordinates of the two endpoints, which is done by adding them together and dividing by 2.

[7 + 31]/2 = 38

38/2 = 19

And 19, of course, is the coordinate of midpoint, V.

Rays

Geometric rays are like the sun's rays. They have a beginning point (or endpoint), and they go on and on without end in a single direction. The sun's rays don't have names (as far as we know). A geometric ray is named by its endpoint and any other point on it.

Above is [??] (read ray DE). The arrow above the letters not only indicates that the figure is a ray. It also indicates the direction in which the ray is pointing. The arrow's head is over the non-endpoint.

This is ray FG. It may be written [??], or [??].

Work Problems

Use these problems to give yourself additional practice.

1. Is the preceding figure more accurately named [bar.MN] or [bar.NM]?

2. PR is 8, and PQ is 22. What is the length of [bar.RQ]?

3. Find the midpoint of [bar.AE].

4. Find the midpoint of [bar.AK].

5. Name the ray here.

6. Name the ray here.

Worked Solutions

1. Both names are suitable. A segment does not have direction, so the order in which its endpoints are named is not relevant.

2. 14 By the Segment Addition Postulate,

PR + RQ = PQ Then 8 + RQ = 22 So RQ = 22 - 8 RQ = 14

3. C [bar.AE]'s endpoints are A and E with coordinates of 6 and 30. That means the length

AE = 30 - 6 AE = 24

The midpoint is half of 24, or 12 from either endpoint. 12 from the starting coordinate of 6 is 18, the coordinate of point ITLITL.

4. E This is essentially the same problem as 3, but we'll use the alternate method to solve it. Let's average the endpoints' coordinates.

First, add them together: 54 + 6 = 60

Then, divide by 2: 60/2 = 30

30 is the coordinate of point E.

5. [??] or [??] The only thing that matters in the naming of a ray is which end has the endpoint and which point is elsewhere on the ray. The endpoint will be under the blunt end of the arrow, and the point that is elsewhere on the ray goes under the arrowhead.

6. [??] or [??] See the explanation for problem 5.

Angles and Angle Pairs

Angles are as important as line segments when it comes to forming geometric figures. Without them, there would be no plane figures, with the possible exception of circles.

Forming and Naming Angles

An angle is formed by two rays that have a common (or shared) endpoint. The rays form the sides of the angle, and their endpoint forms its vertex. The measurement of the opening of an angle is expressed in degrees. The smallest angle possible practical has a degree measure of 0. Imagine the angle formed by the hands of an analog clock at noon. That is a 0 angle. There is no real limit to the upper number of degrees in an angle, but no unique angle contains more than 359, since an angle of 360 is indistinguishable from one of 0.

Rays [??] and [??] form the sides of this angle, shown by the symbol [angle]DEF, whose vertex is at E.

Here we have two pictures of the same angle.

We need two pictures in order to cover all the different ways there are to name this angle. It may be named by three letters-one from each ray and the vertex, with the name of the vertex always being in the middle. Thus, it is [angle]GHI or [angle]IHG. If there is no chance of ambiguity (more than one angle at the vertex), it may be named by the vertex alone, hence [angle]H. It may also be named by a number or a lowercase letter inside the angle, so it is also [angle]1, on the left, or [angel]x, on the right.

Example Problems

These problems show the answers and solutions.

1. State another name for [angle]3.

Answer: [angle]LPM or [angle]MPL Use the name of one point on each ray with the designation of the vertex in the middle. [angle]P would not do, since there are 5 separate angles at vertex P.

2. What is another name for ?KPL? Answer: [angle]LPK or [angle]2 The first is the reverse of the order of the letters in the question (always legitimate), or second, the number that is written in the opening of that angle. 3. What is another name for the sum of angles 4 and 5? Answer: [angle]OPM or [angle]MPO This wasn't really a fair question, since we haven't yet discussed angle addition, still it is a natural application of the already noted fact that a whole is the sum of its parts. The two angles combined into one are bounded by PM and PO, hence the two choices given.

The Protractor Postulate and Addition of Angles

The Protractor Postulate, Postulate 9, supposes that a point, Z, exists on line XY. Think of all rays with endpoint Z that exist on one side of line XY. Each of those rays may be paired with exactly one number between 0 and 180, as you can see in the preceding figure. The positive difference between two numbers representing two different rays is the degree measure of the angle with those rays as its sides. So, the measure of [angle]VZW (represented m[angle]VZW) = 45 - 30 = 15.

Postulate 10 is the Addition of Angles postulate. Simply put, if [??] lies between [??] and [??], then m[angle]XZT = m[angle]XZS + m[angle]SZT. It's just another statement of the whole equaling the sum of its parts, as already used in the previous problems.

(Continues...)

Excerpted from CliffsNotes Geometry Practice Packby David Alan Herzog Copyright © 2010 by John Wiley & Sons, Ltd. Excerpted by permission.
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Edition details

• Publisher: Cliffs Notes Inc.,U.S.
• Publication date: 2010
• ISBN 10: 0470488697
• ISBN 13: 9780470488690
• Binding: Paperback
• Language: English
• Number of pages: 264