Geometry-Mathematics_Resource_Part_II-Chu.pdf

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Mathematics Resource

Part II of III: Geometry

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GEOMETRY

INTRODUCTION TO LINES, PLANES, AND ANGLES

Point

Line

Ray

Angle

Vertex

Geometry is a special type of mathematic that has fascinated man for centuries. Egyptian

hieroglyphics, Greek sculpture, and the Roman arch all made use of specific shapes and their

properties. Geometry is important not only as a tool in construction and as a component of the

appearance of the natural world, but also as a branch of mathematics that requires us to make

logical constructs to deduce what we know.

In the study of geometry, most terms are rigorously defined and used to convey

specific conditions and characteristics. A few terms, however, exist primarily as

concepts with no strict mathematical definition. The point is the first of these

ideas. A point is represented on paper as a dot. It has no actual size; it simply

represents a unique place. To the right are shown three points, named C, H, and

U (capital letters are conventionally used to label and represent points in text).

The next geometric idea is that of the line . A line is a one-dimensional object

that extends infinitely in both directions. It contains points and is represented

as a line on paper with arrows at both ends to indicate that it does indeed

extend indefinitely. Lines are named either with two points that lie on the line

or with a script letter. CH , HC , and m all refer to the same line in the

diagram at left.

Similar to the line is the ray . Definitions for rays vary, but in general, a ray can be

thought of as being similar to a line, but extendi ng infi nitely in only ONE direction.

It has an endpoint and the n extends infinitely in any one direction away from that

endpoint. You might want to think of this endpoint as a Ðbeginning-point.Ñ Rays

are named similarly to lines, but their endpoints must be listed first. Because of

this, CH and HC do not refer to the same ray. CH is shown at right. HC would

point the other direction.

Last in our introduction to basic geometry is the angle . The definition for an

angle remains pretty consistent from textbook to textbook; it is the figure

formed by two rays with a common endpoint, known as the angleÓs vertex .

An angle is named in one of three ways: (1) An angle can be named with the

vertex point if it is the only angle with that vertex. (2) An angle can be named

with a number that is written inside the angle. (3) Most commonly, an angle is

named with three points, the center point representing the vertex. In the

drawing at left, ∠ 1, ∠ C, ∠ UCH, and ∠ HCU all refer to the same angle. In

the drawing at right, ∠ 5 and ∠ BAC refer to the same angle while ∠ 6 and

∠ CAD refer to the same angle. ∠ BAD then refers to an entirely different

angle. Note that ∠ A cannot be used to refer to an angle in the diagram at

right because there are several angles that have vertex A. There would be

no way of knowing which you meant.

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MORE ON LINES AND RAYS, AND A BIT ON PLANES

Distance

Ruler Postulate

Between

Collinear

Line Segment

Midpoint

Congruent

Plane

Coplanar

Now that we have defined and discussed lines, rays, angles, points, and vertices [plural of vertex],

we can begin to set up a framework for geometry. Between any two points, there must be a positive

distance . Even better, between any two points A and B on AB , we can write the distance as AB or

BA. The Ruler Postulate then states that we can set up a one-to-one correspondence between

positive numbers and line distances. In case that sounds like Greek 1 , it means that all distances can

have numbers assigned to the m, and weÓll never run out of numbers in case we have a new distance

that is so-and-so times as long, or only 75% as long, etc. Think of the longest distance you can.

Maybe a million million miles? Well, now add one. A million million and one miles is indeed longer

than a million million mi les. You can keep doing this forever.

Next, we say that a point B is between two others A and C if all three points are collinear (lying on

the same line) and

AB =

. For example, given MN as shown, (remember that we could

also call it

, or a whole slew of other names 2 ),

We might be able to assign distances such that MR = 3, RT = 15, and TN = 13. Intuitively, RN

would then equal 28 (because 15 + 13 = 28), and MT would equal 18 (because 3 + 15 = 18). (Can

you find the distance MN?) Also note that R is between M and T, T is between R and N, T is

between M and N, and R is between M and N. There are many true statements we could make

concerning the betweenness properties on this line. In addition, M, R, T, and N are four collinear

points.

Example :

Fo ur p oints A, B, C, and D are c ollin ear and lie on the line in that order. If B is the midpoint

of AD and C is the midpoint of BD , what is AD in terms of CD?

Solution :

DonÓt just try to think it through; draw it out. The drawing is so me what similar to the one

above with points M, R, T, and N. Since C is the midpoint of BD ,

BD ⋅

and since B

is the midpoint of AD ,

⋅

Earlier, we came to a consensus 3 concerning what exactly lines and rays are. Now we explicitly

define a new concept: the line segment . A line segment consists of two points on a line, along with

all points between them. (Note that geometry is a very logical and tiered branch of mathematics; we

had to define what it meant to be between before we could use that word in a definition.) The

notation for line segments is similar to that for lines, but there are now no arrows over the ends of

1 As my high school physics teacher, Mr. Atman, pointed out to me, Ðthe farther you go in your schooling,

the more important dead Greek guys will become.Ñ ItÓs true. WeÓll get to Pythagoras soon, and

remember that we are studying Euclidean geometry.

2 Finally, a time comes when name-calling is a good thing! ItÓs amazing the things we get to do in

mathematics. Go ahead and have fun with it; I promise the line wonÓt mind.

3 Okay, technically, only I came to the consensus. Pretend you had some input on it.

= 2

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the bar. We can also say that the midpoint B of a line segment AC is the point that divides AC in

half; in other words, the point where AB = BC. It should make sense that every line segment has a

uni que mi dpoint dividing the original segment into two sma ller congruent segments. Two things in

geometry are congruent if they are exactly the same size: two line segments are congruent if their

lengths are equal, two angles are congruent if they have the same angle measure (more on that

later), two figures are congruent if all their sides and angles are eq ual, etc. We writ e co ngruence

almost like the Ð=Ñ sign, but with a squiggly line on top. To say AB is congrue nt to CD , we write

The final concept in this little section is the idea of the plane . Most people learn that a plane is a

surface extending infinitely in two di mensions. The pa per you are reading is a piece of a plane

(provided it is not curled up and wrinkled). The geometric definition of a plane that many books offer

is Ða surface for which containment of two points A and B also implies containment of all points

between A and B along AB .Ñ Essentially, this is a convoluted but very precise way of saying that a

plane must be completely flat and infinitely extended; otherwise, the line AB would extend beyond

its edge or float above or below it. 4

We have now discussed several basic terms, but there remain several key concepts concerning

lines and planes that deserve emphasis. First is the idea that two distinct points determine a unique

line. Think about this: if you take two points A and B anywhere in space, the one and only line

through both A and B is AB (which we could also call BA , of course). Second is the idea that two

distinct lines intersect in at most one point. This is easy to conceptualize: to say that two lines are

distinct is to say that they are not the same line, and two different lines must intersect each other

once or not at all. Lastly, consider the idea that three non-collinear points determine a unique plane;

related to that, also consider the concept that if two distinct lines intersect, there is exactly one plane

containing both lines. Take three non-collinear points anywhere in space, and try to conceive a

plane containing all three of them; there should only be one possible. (Do you understand why the

points must be non-collinear? There are an infinite number of planes containing any given single

line.) Similarly, if two distinct lines intersect, take the point of intersection, along with a distinct point

from each lineÏthis gives three non-collinear points, and a uni que pla ne is determined!

Any plane can be named with either a script letter or three non-collinear points. Here, this plane

could be called pla ne n , plane AEB, or even plane

CEA. We could not , however, call it plane BEC

because B, E, and C are collinear points and do not

uni que ly determi ne one pla ne. Note also now that

BC and AE intersect in only one point, E. In this

drawing 5 , we say that A, B, C, and E are coplanar

points because they all reside on the same plane. It

should make sense to you that any three points must

be coplanar, but four points are only sometimes coplanar. The fourth point may be ÐaboveÑ or

ÐbelowÑ the plane created by the other three 6 .

4 Many formal definitions in mathematics seem very convoluted upon first (or even tenth) glance, but itÓs

the fine points that make these definitions useful to mathematicians.

5 My drawings, IÓm afraid, are rather subpar. If something (as in this case) is supposed to have depth or

more than two-dimensions, youÓll just have to pretend. Pretend really hard.

6 Why do you think that stools with four legs sometimes wobble while stools with three legs never do? ItÓs

because a stool with three legs can always have the ends of its legs match the plane of the floor while if

four legs arenÓt all the same length, they wonÓt be able to all stay coplanar on the floor. Cool, huh?

- Craig

AB ≅ .

GEOMETRY RESOURCE

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EVEN MORE ON LINES, BUT FIRST A BIT ON ANGLES Î A VERY BIG BIT

Protractor Postulate Straight Angle

Angle measure

Right Angle

Acute Angle

Obtuse Angle

Complementary

Supplementary

Adjacent Angles

Angle Bisector

Opposite Rays

Vertical Angles

Perpendicular

Parallel lines

Skew lines

Transversal

Corresponding

Alternate Interior

Just two pages ago, we discussed the Ruler Postulate, which says, in a nutshell, that arbitrary

lengths can be assigned to line segments as long as they maintain correspondence with the positive

numbers. Now, we discuss the Protractor Postulate , which states in a similar fashion that arbitrary

measures can be assigned to angles , with 180 ° representing a straight angle and 360 ° equali ng a

full rotation. 7 The degree measure assigned to an angle is called the angle measure , and the

measure of ∠ 1 is written Ðm ∠ 1.Ñ While a straight angle is 180 ° , a right angle is an angle whose

measure is 90 ° . Keep in mind that on diagrams, straight angles can be assumed when we see

straight lines, but right angles conventionally cannot be assumed unless we see a little box in the

angle. An acute angle is an angle whose measure is less than 90 ° , and an obtuse angle is an

angle whose measure is between 90 ° and 180 ° . Two angles whose measures add to 90 ° are then

said to be complementary angles , and two angles whose measures add to 180 ° are said to be

supplementary angles . Adjacent angles are angles that share both a vertex and a ray. An angle

bisector is a ray that divides an angle into two smaller congruent angles. This laundry list of terms

is summed up in the picture below.

• ∠ ABC is a straight angle. Notice it is flat.

• ∠ ABD is a right angle. (We cannot assume

a right angle on drawings, but the little box

is a symbol indicating that ∠ 1 is a right

angle. Whenever you see a box, feel

confident you are dealing with a right

angle.)

• ∠ ABE is an obtuse angleÏits measure is

greater than 90 ° .

• ∠ 2 and ∠ 3 are acute anglesÏthey

measure less than 90 ° .

• ∠ CBD is a right angle. (Because ∠ 1 is a right angle, ∠ ABC is a straight angle, and 180-

90=90.)

• ∠ 2 and ∠ 3 are complementary. Their measures add to 90 ° since ∠ CBD is a right angle.

• ∠ ABD and ∠ CBD are supplementary. They combine to form straight angle ∠ ABC, which

measures 180 ° .

• ∠ ABE and ∠ 3 are supplementary. They, too, combine to form straight angle ∠ ABC, which

measures 180 ° .

• ∠ 1 and ∠ 2 are adjacent because they share both a vertex and a ray, as are ∠ ABE and ∠ 3,

∠ 1 and ∠ DBC, and ∠ 2 and ∠ 3.

• ∠ 1 and ∠ 3 are not adjacent, however; they share vertex B but no commo n ray.

• BD is the angle bisector of ∠ ABC because it divides the straight angle which measures 180 °

into two smaller congruent angles, each of which measures 90 ° .

• Ifm ∠ 2 and m ∠ 3 each happened to measure 45 ° , then BE would be the angle bisector of

∠ DBCÏsplitting ∠ DBC in half.

7 As a student in geometry, I often wondered why 360 ° comprised a rotation. The most widely accepted

theory is that it was arbitrarily defined at some point in mathematical history. This may have been

because 360 is divisible by so many different numbers. Then again, it may well have been arbitrarily

defined by a civilization that used a base-60 number system.

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