# Dr. Math Introduces Geometry: Learning Geometry is Easy! Just ask Dr. Math!

## Paperback(The Math Forum at Drexel University)

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ISBN-13: 9780471225546 Wiley 08/31/2004 The Math Forum at Drexel University 192 7.50(w) x 9.25(h) x 0.41(d) 10 - 14 Years

THE MATH FORUM @ Drexel (www.mathforum.org) is an award-winning Web site and one of the most popular online math resources for students and teachers. The Math Forum offers answers to all kinds of math questions, prepared by a team of math experts. It also keeps archives of previous questions and answers, hosts online communities, and posts several "problems of the week."

#### John Wiley & Sons

ISBN: 0-471-22554-1

#### Chapter One

Introduction to Two-Dimensional (2-D) Geometric Figures

Two-dimensional geometry, coordinate plane geometry, Cartesian geometry, and planar (pronounced PLANE-er) geometry refer to the same thing: the study of geometric forms in the coordinate plane. Do you remember the coordinate plane? It's a grid system in which two numbers tell you the location of a point-the first, x, tells you how far left or right to go from the origin (the center point), and the second number, y, tells you how far up or down to go. The y-axis is vertical and the x-axis is horizontal (like the horizon).

You'll see a lot more of the coordinate plane in geometry, but sometimes all that matters is knowing that a figure is in the plane or two-dimensional without knowing a precise address for it. This part will introduce you to some of the most common figures in two-dimensional geometry and give you some names for their parts and ways to work with them.

In this part, Dr. Math explains

points, lines, and planes

angles

triangles

1 Points, Lines, and Planes

Points, lines, and planes correspond to talking about no dimensions, one dimension, and two dimensions in the coordinate plane. A line is one-dimensional, since one number, the distance from zero, tells you where you are. A plane is two-dimensional, since you need x and y to locate apoint. A point is dimensionless. It consists only of location, so it's only possible to be one place if you're on a point-you don't need any extra numbers to tell you where you are. Points, lines, and planes are the foundations of the whole system of geometry.

But point, line, and plane are all undefined terms. How can that be? Well, any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular!

Undefined Geometry Terms

Dear Dr. Math,

I know that they call point, line, and plane the undefined terms of geometry, but is there a way to give those terms a definition? I've been thinking, could a line be defined as the joining of two rays going in separate directions? I've never really thought that anything couldn't have a definition, so is it possible for any of these geometric terms to be defined?

Yours truly, Leon

Dear Leon,

Your definition would require us to first define "ray" and "direction." Can you do that without reference to "point," "line," and "plane"?

Think of it this way: math is a huge building, in which each part is built by a logical chain of reasoning upon other parts below it. What is the foundation? What is everything else built on?

There must be some lowest level that is not based on anything else; otherwise, the whole thing is circular and never really starts anywhere. The undefined terms are part of that foundation, along with rules that tell us how to prove things are true. The goal of mathematicians has not been to make math entirely self-contained, with no undefined terms, but to minimize the number of definitions so that we have to accept only a few basics, and from there we will discover all of math to be well defined. Also, the goal is to make those terms obvious so that we have no trouble accepting them, even though we can't formally prove their existence.

To put it another way, these terms do have a definition in human terms-that is, we can easily understand what they mean. They simply don't have a mathematical definition in the sense of depending only on other previously defined terms. -Dr. Math, The Math Forum

What Is a Point?

Dear Dr. Math,

Yours truly, Lorraine

Dear Lorraine,

The word "point" is undefined in geometry. But it is pretty easy for us to describe a point, even though it can't be defined. A point is an entity that has only one characteristic: its position. A point has no size, color, smell, or feel. When we talk about points, we are referring to one specific location.

For example, along a number line the number 2 exists at just one point. Points are infinitely small, which means the point at 2 is different from the point at 2.000000001. Here's a picture of a number line:

If you want to distinguish one place along a number line, you "point" at it. You label that place with the corresponding number and refer to it with that number.

Now, how do you distinguish a location in two-dimensional space (e.g., a sheet of paper)? Imagine that we have two number lines: one horizontal and the other vertical. We are pointing at a place p:

How do we describe where the point p is? We can't just say p is at 2 because we don't know which number line that refers to. Is it at 2 along the horizontal number line or the vertical one?

To describe where p is, you must talk about where it is both horizontally and vertically. So, you can say

p is at 2 horizontally and 1 vertically

However, this is a mouthful. Because describing points in two dimensions is really useful, we have defined some conventions to make life easier. We call the horizontal number line the x-axis and the vertical number line the y-axis. The convention for talking about points in two dimensions is to write

(position along x-axis, position along y-axis)

Therefore,

p is at (2, 1)

Points in two dimensions can be described by any pair of numbers. For example, (4, 5), (6.23432, 3.14), and (-12, 4) are all points. -Dr. Math, The Math Forum

Rays, Line Segments, and Lines

Dear Dr. Math,

I need to know what a ray, a line segment, and a line are.

Sincerely, Leon

Dear Leon,

In geometry, you can think of a line just like a normal straight line, with a couple of special features. The things that make a line in geometry different from a line in any other context-for example, art class-are that it goes on forever in both directions, it's perfectly straight, and it's not thick.

Mathematicians say that their lines have zero thickness, which is pretty hard to imagine. When we draw lines on paper, they always have at least a little bit of width. But when we study lines in geometry, we think of them as having no width at all.

Here's how a lot of people draw lines on paper. The arrows at the ends mean that the line continues forever in both directions:

Rays and line segments are a lot like lines. A ray is like a line, except that it only goes on forever in one direction. So it starts at one point and goes on forever in some direction. You can think of the light coming from the sun as an example of a ray: the starting point is at the sun, and the light goes on forever away from the sun.

Here's how we draw rays:

A line segment is a little chunk of a line. It starts at one point, goes for a while, and ends at another point. We draw it like this:

Sometimes we like to attach little dots to represent the endpoints of rays and line segments like this: -Dr. Math, The Math Forum

2 Angles

There are angles all around us-between the hands on a clock, the opening created by a door, even the joints of your body. Any time two lines or line segments or rays intersect, they make angles.

What makes one angle different from another? Angles differ in how far open their "jaws" are. If you think of opening an angle starting with two line segments on top of each other, you could open it a little bit, or a pretty big amount, or a whole lot; you could bend it back on itself until the line segments are almost on top of each other again. We often measure angles in degrees to describe how far open the angles are.

In this section, we'll talk about the different kinds of angles and the ways we measure them.

What Is a Vertex?

Dear Dr. Math

What does vertex mean?

Sincerely, Lorraine

Dear Lorraine,

A vertex is the point at which two rays of an angle or two sides of a polygon meet. Vertices (pronounced VER-tih-seez) is the plural of vertex.

A triangle has three vertices. -Dr. Math, The Math Forum

Types of Angles: Acute, Right, Obtuse, and Reflex

Dear Dr. Math,

How can I remember what the types of angles mean-for example, acute angle or right angle?

Yours truly, Leon

Dear Leon,

There are three main types of angles. We tell them apart by how big they are. (Angles are often measured in degrees; there are 360 degrees in a circle.) Here they are:

We can start with the right angle: a right angle measures exactly 90 degrees. It's called a right angle because it stands upright. Just remember it's an upright angle.

Next is the acute angle. Acute angles measure less than 90 degrees. The word "acute" comes from a word that means "sharp." Remember that a sharp pencil or a sharp knife has an acute angle at its point or blade. An acute pain is a sharp pain. Acupuncture uses sharp needles. And, if all else fails, you can remember that an acute angle can cut you!

Finally, we have the wide-open obtuse angles, which measure between 90 and 180 degrees. The word "obtuse" comes from a Latin verb meaning "to make blunt or dull." If a person isn't very sharp (doesn't have an acute intelligence), he may be called obtuse. If that doesn't stick in your mind, just remember that if it isn't right or acute, it's obtuse.

I should mention a fourth kind of angle: the reflex angle. A reflex angle is the other side of any other type of angle. Reflex angles measure more than 180 degress. For example, in this diagram, the angle labeled A is the reflex angle. (The other angle in the diagram is obtuse.)

One meaning of reflex is "to bend back"; and the angle kind of looks bent back, like an elbow bent too far. Actually, some people can make a reflex angle with their elbow, and some can't. Can you?

I hope the names are memorable for you now. -Dr. Math, The Math Forum

Complementary and Supplementary Angles

Dear Dr. Math,

In class we're studying complements and supplements of angles. I do not understand any of the terminology behind the problems. Today we took a test, and one of the questions was to find the complement of this angle, c degrees, and I didn't even know where to begin. Another was to find the degrees in the third angle in an isosceles triangle, x degrees, x - 10, or something like that. Can you explain this a little better?

Sincerely, Lorraine

Dear Lorraine,

Part of the problem here is that the names "complement" and "supplement" are kind of confusing, since the literal meanings of these words aren't different enough for us to know which is which, other than by memorizing them.

What are complements and supplements?

If you place two angles next to each other so that they add up to 90 degrees, we say that the angles are complements.

If you place two angles next to each other so that they add up to 180 degrees, we say that the angles are supplements.

Here are some examples of complements and supplements:

Complements Supplements

30 and 60 degrees 30 and 150 degrees 2 and 88 degrees 2 and 178 degrees 14 and 76 degrees 14 and 166 degrees

So what you need to remember is which one adds up to 90 degrees and which one adds up to 180 degrees.

How can you keep them straight? The person who runs the Math Forum's Geometry Problem of the Week tells me that she remembers them this way: c comes before s, and 90 comes before 180. It's the best idea I've heard so far.

If you know that two angles are complements or supplements, you can figure out one given the other. How? Well, if they're supplements, you know that they have to add up to 180:

this + that = 180

So it must be true that

this = 180 - that and

that = 180 - this

You can do the corresponding calculations for complements using 90 degrees instead of 180 degrees.

So whenever you see the phrase "the supplement of (some angle)°," you can immediately translate it to "180° - (the angle)°." When you have a value for the angle, you end up with something like

the supplement of 26° = (180° - 26°)

which you can just simplify to get a single number. But if you only have a variable like x, or an expression for the angle, like x - 10, then you just have to deal with that by substituting the variable or the expression in the equation. For example:

the supplement of (x° - 10°) = [180° - (x° - 10°)]

Note that you have to put the expression in parentheses (or brackets), or you can end up with the wrong thing. In this case,

[180° - (x° - 10°)] is not the same as (180° - x° - 10°) (180° - x° + 10°) (180° - 10° - x°) (190° - x°) (170° - x°)

Why should you care about complements and supplements?

Well, in geometry you're constantly dividing things into triangles in order to make them easier to work with. And in every triangle, the measures of the interior angles add up to 180 degrees. So if you know two angles, the third is the supplement of the sum of the other two.

The nicest kind of triangle to work with is a right triangle. In a right triangle, you have one right angle and two other angles. Since they all have to add up to 180 degrees, and since the right angle takes up 90 of those degrees, the other two angles must add up to 90 degrees. So if you know one of the acute angles in a right triangle, the other is just the complement of that angle. -Dr. Math, The Math Forum

RX ORDER OF OPERATIONS

In case you've forgotten, here's a quick review of the correct order of operations for any expression:

1. Parentheses or brackets

2. Exponents

3. Multiplication and division (left to right)

4. Addition and subtraction (left to right)

Alternate and Corresponding Angles

Dear Dr. Math,

Please explain alternate and corresponding angles.

Sincerely, Leon

Dear Leon,

Let's first look at a diagram that we can refer to when we define corresponding angles and alternate angles:

There are a lot of numbers in this diagram! Don't worry, though-we'll figure out what everything means.

Assume that the two horizontal lines are parallel (that means they have the same slope and never intersect). The diagonal is called a transversal, and as you can see, the intersection of the transversal with the horizontal lines makes lots of different angles. I labeled these angles 1 through 8. Whenever you have a setup like this in which you have two parallel lines and a transversal intersecting them, you can think about corresponding angles and alternate angles.

Look at the diagram. Do you see how we could easily split the angles into two groups? Angles 1, 2, 3, and 4 would be the first group-they are the angles the transversal makes with the higher horizontal line. Angles 5, 6, 7, and 8 would be the second group-they are the angles the transversal makes with the lower horizontal line.

Can you see how the bottom set of four angles looks a lot like the top set of four angles? We say that two angles are corresponding angles if they occupy corresponding positions in the two sets of angles. For example, 1 and 5 are corresponding angles because they are both in the top left position: 1 is in the top left corner of the set of angles {1, 2, 3, 4}, while 5 is in the top left corner of the set of angles {5, 6, 7, 8}.

Similarly, 3 and 7 are corresponding angles. There are two more pairs of corresponding angles in the diagram. Can you find them?

One neat and helpful fact about corresponding angles is that they are always equal. Can you see why? (Think about the way the nonparallel line intersects the parallel lines.)

Let's move on to alternate angles.

(Continues...)

Excerpted from Dr. Math Introduces Geometry Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Acknowledgments.

Introduction.

PART 1: INTRODUCTION TO TWO-DIMENSIONAL (2-D) GEOMETRICFIGURES.

1. Points, Lines, Planes.

2. Angles.

3. Triangles.

Resources on the Web.

PART 2: AREAS AND PERIMETERS OF TWO-DIMENSIONAL (2-D) GEOMETRICFIGURES.

1. Area and Perimeter.

2. Units of Area.

3. Area and Perimeters of Parallelograms and Trapezoids.

Resources on the Web.

PART 3: CIRCLES AND PI.

1. Pi, Circle Parts, and Circle Measurements.

Resources on the Web.

PART 4: INTRODUCTION TO THREE-DIMENSIONAL (3-D) GEOMETRICFIGURES.

1. Polyhedra.

2. Platonic Solids.

3. Surface Area.

4. Volume.

5. Nets of Solids.

Resources on the Web.

PART 5: SYMMETRY.

1. Rigid Motions: Rotation, Reflection, Translation, GlideReflection.

2. Symmetries.

3. Lines of Symmetry.

4. Tessellations.

Resources on the Web.

Appendix: Geometric Figures.

Glossary.

Index.

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