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Geometry

Points, Lines, Planes & Angles

Part 1

www.njctl.org

2014-09-05

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Table of ContentsIntroduction to Geometry

click on the topic to go to that section

Points and LinesPlanes Congruence, Distance and LengthConstructions and Loci

Part 1

Part 2AnglesCongruent AnglesAngles & Angle Addition PostulateProtractorsSpecial Angle PairsProofs Special Angles

Angle Bisectors & ConstructionsLocus & Angle ConstructionsAngle Bisectors

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http://www.njctl.org




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Introduction to

Geometry

Return to Table of Contents

Slide 4 / 206

The Origin of Geometry

About 10,000 years ago much of North Africa was fertile farmland.

The area around the Nile river was too marshy for agriculture, so it was sparsely populated.

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The Origin of GeometryBut over thousands of years the climate changed, and most of North

African became desert.

The banks of the Nile became prime farmland.

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The Origin of Geometry

The land along the Nile became crowded with people.

Farming was done on the land near the river because it had:

· Water for irrigation

· Fertile soil due to annual flooding, which deposited silt from upriver.

But, since the land flooded each year, how could they keep track of who owned which land?

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About 4000 years ago an Egyptian pharaoh, Sesostris, is said to have invented geometry in order to keep track of the land and tax it's owners.

Reestablishing land ownership after each annual flood required a practical geometry.

"Geo" means Earth and "metria" means measure, so geometry meant to measure land.

Egyptian Geometry

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You know more geometry than the Egyptians knew 4000 years ago, so let's do a lab to see how you would solve this problem.

Land Boundaries Lab

You'll work in groups and each group will solve this problem

before we move on to how the Greek's built on the Egyptian

solution.

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Before the annual flood of the Nile three plots of land might be as shown.

The orange dots are to indicate stakes that were placed above the flood level.

The stakes would remain in the same location from year to year.

A

Plot 1

C

B

D

E

Plot 3

Plot 2

Pre- Flood Boundary Map

Land Boundaries Lab

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Before flooding, three plots of land might be look like these.

Land Boundaries Lab

A

Plot 1

C

B

D

E

Plot 3

Plot 2

Pre- Flood Boundary Map

A

C

B

D

E

Post-Flood Map of River and Markers

Afterwards, only the stakes above the flood level remained, and the

river had moved in its course.

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The pharaoh had to:

· Reestablish new boundaries so farmers knew which land to farm.

· Adjust the taxes to match the new amount of land owned.

Land Boundaries Lab

The Egyptians only had stakes and rope, you only have tape and string.

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After the flood, the pharaoh would send out geometers with ropes that had been used to measure each plot of land in prior years.

How did they do it?

(You can't use the edges of the paper or rulers because these were open fields of great size.)

Land Boundaries Lab

A

C

B

D

E

Post-Flood Map of River and Markers

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Egyptian mathematics was very practical. What practical applications do you think the Egyptians used mathematics for?

They did not develop abstract mathematics. That was left to the Greeks, who built upon what they had learned from the Egyptians, Babylonians and others.

Egyptian GeometryTe

ache

r Not

es

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Greek Geometry

The Greeks developed an approach to thinking about these earth measures that allowed them to be generalized.

They kept their assumptions to the minimum, and showed how all else followed from those assumptions.

Those assumptions are called definitions, postulates and axioms.

That analytical thinking became the logic that allows us to not only measure land, but also measure the validity of ideas.

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Euclidean Geometry

Euclid's book, The Elements, summarized the results of Greek geometry: Euclidean Geometry.

Euclidean geometry is the basis of much of western mathematics, philosophy and science.

It also represents a great place to learn that type of thinking.

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Euclidean Geometry

Euclidean Geometry dates prior to 400 BC.

That makes it about 1000 years older than algebra, and about 2000 years older than calculus.

The fact that it is still taught in much the way it was more than 2000 years ago tells us what about Euclid's ideas?

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Euclidean Geometry

This statement was posted above Plato's Academy, in ancient Athens, about 2500 years ago.

This renaissance painting by Raphael depicts that academy.

"Let none who are ignorant of geometry enter here."

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Euclidean Geometry

When the Roman empire declined, and then fell, about 1800 years ago, most of the writings of Greek civilization were lost.

This included most of the plays, histories, philosophical, scientific and mathematical works of that era, including The Elements by Euclid.

These works were not purposely destroyed, but deteriorated with age as there was no central government to maintain them.

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Euclidean Geometry

Euclidean Geometry was lost to Europe for a 1000 years.

But, it continued to be used and developed in the Islamic world.

In the 1400's, these ideas were reintroduced to Europe.

These, and other rediscovered works, led to the European Renaissance, which lasted several centuries, beginning in the 1400's.

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Euclidean Geometry

Much of the thinking of modern science and mathematics developed from the rediscovery of Euclid's Elements.

The thinking that underlies Euclidean Geometry has held up very well.

Many still believe it is the best introduction to analytical thinking.

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Euclidean GeometryAbout 100 years ago, Charles Dodgson, the Oxford geometer who wrote Alice in Wonderland, under the name Lewis Carroll, argued Euclid was still the best way to understand mathematical thinking.

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Euclidean Geometry

Geometry is used directly in many tasks such as measuring lengths, areas and volumes; surveying land, designing optics, etc.

Geometry underlies much of science, technology, engineering and mathematics (STEM).

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Euclidean Geometry

This course will use the basic thinking developed by Euclid.

We will attempt to make clear and distinguish between:

· What we have assumed to be true, and cannot prove· What follows from what we have previously assumed or proven

That is the reasoning that makes geometric thinking so valuable. Always question every idea that's presented, that's what Euclid

and those who invented geometry would have wanted.

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Euclidean Geometry

This also represents a path to logical thinking, which British philosopher Bertrand Russell showed is identical to mathematical thinking.

Click on the image to watch a short video of Bertrand Russell's message to the future which was filmed in 1959.

Did you hear anything that sounded familiar?

What was it?

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Euclidean Geometry

Euclid's assumptions are axioms, postulates and definitions.

You won't be expected to memorize them, but to use them to develop further understanding.

Major ideas which are proven are called Theorems.

Ideas that easily follow from a theorem are called Corollaries.

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Euclidean Geometry

The five axioms are very general, apply to the entire course, and do not depend on the definitions or postulates, so we'll review them in this unit.

The postulates and definitions are related to specific topics, so we will introduce them as required.

Also, additional modern terms which you will need to know will be introduced as needed.

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Euclid called his axioms "Common Understandings."

They seem so obvious to us now, and to him then, that the fact that he wrote them down as his assumptions reflects how carefully he wanted to make clear his thinking.

He didn't want to assume even the most obvious understandings without indicating that he was doing just that.

Euclid's Axioms (Common Understandings)

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This careful rigor is what led to this approach changing the world.

Great breakthroughs in science, mathematics, engineering, business, etc. are made by people who question what seems obviously true...but turns out to not always be true.

Without recognizing the assumptions you are making, you're not able to question them...and, sometimes, not able to move beyond them.


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Things which are equal to the same thing are also equal to one another.

Euclid's First Axiom

For example:

if I know that Tom and Bob are the same height, and I know that Bob and Sarah are the same height...what other conclusion can I come to?

Tom Bob Sarah

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If equals are added to equals, the whole are equal.

Euclid's Second Axiom

For example,

if you and I each have the same amount of money, let's say $20, and we each earn the same additional amount, let's say $2,

then we still each have the same total amount of money as each other, in this case $22.

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If equals be subtracted from equals, the remainders are equal.

Euclid's Third Axiom

This is just like the second axiom.

Come up with an example on your own. Look back at the second axiom if you need a hint.

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Things which coincide with one another are equal to one another.

Euclid's Fourth Axiom

For example,

if I lay two pieces of wood side by side and both ends and all the points in between line up, I would say they have equal lengths.

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The whole is greater than the part.

Euclid's Fifth Axiom

For example,

if an object is made up of more than one part,

then the object has to be larger than any of those parts.

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First Axiom: Things which are equal to the same thing are also equal to one another. Second Axiom: If equals are added to equals, the whole are equal.

Third Axiom: If equals be subtracted from equals, the remainders are equal.

Fourth Axiom: Things which coincide with one another are equal to one another.

Fifth Axiom: The whole is greater than the part.


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Points and Lines


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Definitions

Definitions are words or terms that have an agreed upon meaning; they cannot be derived or proven.

The definitions used in geometry are idealizations, they do not physically exist.

When we draw objects based on these definitions, that is just to help visualize them. However, imaginary geometric objects can be

used to develop ideas that can then be made into real objects.

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Points

A point is infinitely small.

It cannot be divided into smaller parts.

It is a location in space, without dimensions.

It has no length, width or height.

Definition 1: A point is that which has no part.

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Points

Definition 1: A point is that which has no part.

Look at this dot. Why can it not be considered a point?Discuss your answer with a partner.

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Points

A point is represented by a dot. The dot drawn on a page has dimensions, but the point it represents does not.

A point can be imagined, but not drawn.

Only the position of the point is shown by the dot.

Points are usually labeled with a capital letter (e.g. A, B, C).

A B

C

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Lines

A line is defined to have length, but no width or height.

The line drawn on a page has width, but the idea of a line does not.

Definition 2: A line is breadthless length.

Lines can be thought of as an infinite number of points with no space between them.

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Lines

A line consists of an infinite number of points laid side by side, so at either end of a line are points.

These are called endpoints.

Definition 3: The ends of a line are points.

Even though this is how we correctly depict a line with endpoints, why is is not accurate?

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Lines

Definition 4. A straight line is a line which lies evenly with the points on itself.

In a straight line the points lie next to one another without bending or turning in any direction.

While a line can follow any path, in this course we will use the term "line" to mean a straight line, unless otherwise indicated.

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First Postulate: To draw a line from any point to any point.

Lines

This postulate indicates that given any two points, it is possible to draw a line between them.

Aside from letting us connect two points with a line, it also allows us to extend any line as far as we choose since points could be

located at any point in space.

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Lines

Second Postulate: To produce a finite straight line continuously in a straight line.

This postulate indicates that the line drawn between any two points can be a straight line.

This allows the use of a straight edge to draw lines.

A straight edge is a ruler without markings.

Note: Any object with a straight edge can be used.

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Line Segments

Using these definitions and postulates we can first draw two points (the endpoints) and then draw a straight line between them using a straight edge.

A line drawn in this way is called a line segment.

It has finite length, a beginning and an end.

At each end of the segment there is an endpoint, as shown below

A Bendpoint endpoint

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Naming Line Segments

For instance, and are different names for the same segment. AB BA

A line segment is named by its two endpoints.

The order of the endpoints doesn't matter.


AB or BA

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A straight line, which extends to infinity in both directions, can be created by extending a line segment in both directions.

This is allowed by our definitions and postulates by imagining connecting each endpoint of the segment to other points that lie beyond it, in both directions.

Lines

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A B

In this example, line Segment AB is extended in both directions to create Line AB.

Lines


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DE

A line is named by using any two points on it OR by using a single lower-case letter.

Arrowheads in the symbol above the points in the name of the line show that the line continues without end in opposite directions.

Naming Lines

D

F

E

a

DF EF

FEED FD a

Here are 7 valid names for this line.

When using two points to name a line, their order doesn't matter since the line goes in both directions.

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Give 7 different names for this line.

Example

U

W

V

b

Ans

wer

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Collinear points are points which fall on the same line.

Which of these points are collinear with the drawn line?

Collinear Points

D

F

E

a

A

B C

Ans

wer

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Is it possible for any two points to not be collinear on at least one line?

Come up with an answer at your table. Remember, only use facts to make your argument!

Collinear PointsA

nsw

er

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1 How many points are needed to define a line?

Ans

wer

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2 Can there be two points which are not collinear on some line?

Yes

No

Ans

wer

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3 Can there be three points which are not collinear on some line?

YesNo

Ans

wer

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Is it possible for two different lines to intersect at more than one point?

Intersecting Lines

A good technique to prove whether this is possible is called either

Argumentum ad absurdum

or

Reductio ad absurdum

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Intersecting Lines

Argumentum ad absurdum

or

Reductio ad absurdum

These are two Latin terms which refer to the same powerful approach, an indirect proof.

First, you assume something is true. Then you see what logically follows from that assumption. If the conclusion is absurd, the assumption was

false, and disproven.

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Intersecting Lines

Let's assume that two different lines can share more than one point and see where that leads us.

Let's name the two points which are shared A and B.

We could connect A and B with a line segment, since we can draw a line segment between any two points.

That segment would overlap both our original lines between A and B, since they are all straight lines and all include A and B.

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Intersecting Lines

We could then extend our Segment AB infinitely in both directions and our new Line AB would overlap our original two lines to infinity in both directions.

If they share all the same points, they are the same lines, just with different names.

But we assumed that the two original lines were different lines sharing two points.

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Intersecting Lines

But we have concluded that they are the same line, not different lines.

It is impossible for them to be both different lines and the same lines.

So, our assumption is proven false and the opposite assumption must be true.

Two different lines cannot share two points.

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Intersecting Lines

Q

T

K

R

S

So, two different lines either:

· Intersect at no points

· Intersect at one point.

F

E

D

C

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4 What is the maximum number of points at which two distinct lines can intersect?

Ans

wer

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5 Which sets of points are collinear on the lines drawn in this diagram?

A

CD

B

A A, D, BB C, D, BC A, D, CD none

Ans

wer

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6 At which point, or points, do the drawn lines intersect?

A A and DB A and CC DD none

A

CD

B

Ans

wer

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Below, the segment AB is extended to infinity, beyond Point B, to create Ray AB.

Rays

A B


A Ray is created by extending a line segment to infinity in just one direction. It has a point at one end, its endpoint, and extends to infinity at the other.

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Naming Rays

When naming a ray the first letter is the point where the ray begins and the second is any other point on the ray.

The order of the letters matters for rays, while it doesn't for lines.

Why do you think the order of the letters matter for rays?

A B

A B

Line AB or Line BA

Ray AB

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Naming Rays

Also, instead of the double-headed arrows which are used for lines, rays are indicated by a single-headed arrow.

The arrow points from the endpoint of the ray to infinity.

A B

A B

AB or BA

AB

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Naming Rays

Segment AB can be extended in either in either direction.

We can extend it at B to get ray AB.

Or, we can extend it at A to get Ray BA.

A BAB

A B

A BBA

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Rays AB and BA are NOT the same. What is the difference between them?

Naming Rays

A BAB

A BBA

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Below, suppose point C is between points A and B.

Rays CA and CB are opposite rays.

Opposite rays are defined as being two rays with a common endpoint that point in opposite directions and form a straight line.

Opposite Rays

A BC

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Recall: Since A, B, and C all lie on the same line, we know they are collinear points.

Similarly, rays are also called collinear if they lie on the same line.

Collinear Rays

A BC

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7 Name a point which is collinear with points G & H.

A

BCDEFGH

C

DG

A

FH

B

E

Ans

wer

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8 Name a point which is collinear with points D & A.

A

BCDEFGH

C

DG

A

FH

B

E

Ans

wer

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9 Name a point which is collinear with points D & E.

A

BCDEFGH

C

DG

A

FH

B

E

Ans

wer

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10 Name a point which is collinear with points C & G.

A

BCDEFGH

C

DG

A

FH

B

E

Ans

wer

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11 Name an opposite ray to Ray MN.

A Ray MQ

B Ray MO

C Ray RO

D Ray PRO

QP

M

TR

N

S

Ans

wer

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12 Name an opposite ray to Ray PS.A Ray MQB Ray MOC Ray POD Ray PR

O

QP

M

TR

N

S

Ans

wer

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13 Name an opposite ray to Ray PM.A Ray MQB Ray MOC Ray POD Ray PR

O

QP

M

TR

N

S

Ans

wer

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14 Rays HE and HF are the same. True

False

D

H

g

P

G

E

F

p

Ans

wer

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15 Rays HE and HP are the same. True

False

D

H

g

P

G

E

F

p

Ans

wer

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16 Lines EH and EF are the same.True

False

D

H

g

P

G

E

F

p

Ans

wer

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17 Line p contains just three points.

True

False

D

H

g

P

G

E

F

p

Ans

wer

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18 Points D, H, and E are collinear.

True

False

D

H

g

P

G

E

F

p

Ans

wer

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19 Points G, D, and H are collinear.

True

False

D

H

g

P

G

E

F

p

Ans

wer

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20 Are ray LJ and ray JL opposite rays?

Yes

No

J

K

L

Ans

wer

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21 Which of the following are opposite rays?

A JK & LK

B JK & LK

C KJ & KL

D JL & KL

J

K

L

Ans

wer

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22 Name the initial point of ray AC.

A

B

C

A

B

C

Ans

wer

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23 Name the initial point of ray BC.

A

B

C

A

B

C

Ans

wer

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Planes


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Planes

A plane is a flat surface that has no thickness or height.

It can extend infinitely in the directions of its length and breadth, just as the lines that lie on it may.

But it has no height at all.

Definition 5: A surface is that which has length and breadth only.

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Planes

Recall that points which fall on the same line are called collinear points.

With that in mind, what do you think points on the same plane are called?

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Planes

Just as the ends of lines are points, the edges of planes are lines.

Definition 6: The edges of a surface are lines.

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Planes

This indicates that the surface of the plane is flat so that lines on the plane will lie flat on it.

Thinking about the definitions of points and lines, exactly how flat do you think a plane is?

Definition 7: A plane surface is a surface which lies evenly with the straight lines on itself.

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As you figured out earlier, coplanar points are points which fall on the same plane.

Coplanar Points and Lines

All of the lines and points shown here are coplanar.

D

F

E

a

A

B C

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Naming Planes

Also, it can be named by the single letter, "Plane R."

Planes can be named by any three points that are not collinear.

This plane can be named "Plane KMN," "Plane LKM," or "Plane KNL."

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Coplanar Points

Coplanar points lie on the same plane.

In this case, Points K, M, and L are coplanar and lie on the indicated plane.

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While points O, K, and L do not lie on the indicated plane, they are coplanar with one another.

Can you imagine a plane in which they are coplanar?

Can you draw it on the image? What could be a name for that plane?

Coplanar Points

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Is it possible for any three points to not be

coplanar with one another?

Try and find 3 points on this diagram which are not

coplanar.

Coplanar Points

Ans

wer

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24 How many points are needed to define a plane?

Ans

wer

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25 Can there be three points which are not coplanar on any plane?

YesNo

Ans

wer

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26 Can there be four points which are not coplaner on any plane?

YesNo

Ans

wer

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What would the intersection of two planes look like?

Hint: the walls and ceiling of this room could represent planes.

Intersecting Planes

Ans

wer

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A B

The intersection of these two planes is shown by Line AB.

Intersecting Planes

Try to imagine how two planes could intersect at a point, or in any other way than a line.

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Various Planes Defined by 3 points

Imagine or shade in Plane BAW in the below drawing.

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Plane BAW

What are the 3 other ways you can name this

same plane?

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Various Planes Defined by 3 points Imagine or shade in Plane AZW in the below drawing.

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Plane AZW


same plane?

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Various Planes Defined by 3 points Draw Plane UYA in the below drawing.

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Plane UYA


same plane?

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Various Planes Defined by 3 points Imagine or draw Plane ABU in the below drawing.

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Various Planes Defined by 3 points Plane ABU


same plane?

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27 Name the point that is not in plane ABC.

A

BCD

A

B

C

D Ans

wer

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28 Name the point that is not in plane DBC.

A

BCD

A

B

C

D Ans

wer

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29 Name two points that are in both indicated planes.ABCD

A

B

C

D

Ans

wer

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30 Name two points that are not on Line BC.ABCD

A

B

C

D

Ans

wer

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31 Line BC does not contain point R. Are points R, B, and C collinear? Draw the situation if it helps.

Yes

No Ans

wer

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32 Plane LMN does not contain point P. Are points P, M, and N coplanar?

Yes

No

Ans

wer

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33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture)

Yes

No

Ans

wer

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34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar?

Yes

No

Ans

wer

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35 Line BA and line DB intersect at Point ____.

ABCDEFGH

Ans

wer

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36 Which group of points are noncoplanar with points A, B, and F on the cube below.

A E, F, B, A

B A, C, G, E

C D, H, G, C

D F, E, G, H Ans

wer

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37 Are lines EF and CD coplanar on the cube below?

Yes

No

Ans

wer

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38 Plane ABC and plane DCG intersect at _____?

A C

B line DC

C Line CG

D they don't intersect

Ans

wer

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39 Planes ABC, GCD, and EGC intersect at _____?

A line GC B point C

C point A

D line AC

Ans

wer

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40 Name another point that is in the same plane as

points E, G, and H.

A

B

C

D

E

F

G

H

Ans

wer

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41 Name a point that is coplanar with points E, F, and C.

A

B

C

D

E

F

G

H

Ans

wer

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42 Intersecting lines are __________ coplanar.

A Always

B Sometimes

C Never

Ans

wer

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43 Two planes ____________ intersect at exactly one point.

A Always

B Sometimes

C Never Ans

wer

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44 A plane can __________ be drawn so that any three points are coplaner

A Always

B Sometimes

C Never

Ans

wer

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45 A plane containing two points of a line __________ contains the entire line.

A Always

B Sometimes

C Never Ans

wer

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46 Four points are ____________ noncoplanar.

A Always

B Sometimes

C Never

Ans

wer

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47 Two lines ________________ meet at more than one point.

A Always

B Sometimes

C Never

Ans

wer

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Congruence, Distance and Length


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Two objects are congruent if they can be moved, by any combination of translation, rotation and reflection, so that every part of each object overlaps.

This is the symbol for congruence:

If a is congruent to b, this would be shown as below:

which is read as "a is congruent to b."

a b

Congruence

Slide 135 / 206

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By this definition, it can be seen that all lines are congruent with one another.

They are all infinitely long, so they have the same length.

If they are rotated so that any two of their points overlap, all of their points will overlap.

Congruence

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Two objects are congruent if they can be moved, by translation, reflection, and/or rotation, so that every point of each object overlaps every point of the other object.

There's no problem rotating line b to overlap line a.

Congruence

a b

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And they are both infinitely long, so they have the same length.

Therefore, they will overlap at every point once they are rotated to overlap at 2 points.

They are congruent.

Congruence

a b

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Would the same be true for any two rays?

Congruence

a b

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Again, all rays are infinitely long, so they have the same length.

And once their vertices and any other point on both rays overlap, all of their points will overlap.

All rays are congruent.

Congruence

a

b

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Would the same be true of all line segments?

Congruence

a b

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If two line segments have different lengths, no matter how I move or rotate them, they will not overlap at every point.

Only segments with the same length are congruent.

Congruence

ab

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While distance and length are related terms, they are also different.

At your table, come up with definitions of Distance and Length which show how they are related and how they are different.

Distance and Length

Distance:

Length:

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Distance is defined to be how far apart one point is from another.

Length is defined to be the distance between the two ends of a line segment.

Since every line segment has a point at each end, these are closely related concepts.

To show congruence of line segments, they must show they have the same length.

Distance and Length

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A B C D E F

Ruler Postulate: Any location along a number line can be paired with a matching number.

This can be used to create a ruler in order to measure lengths and distances.

Distance and Length

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A B C D E F

For instance, we can indicate that on the below number line:

Point C is located at the position of 0.

Point E is located at +7.

Distance and Length

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We can say that points C and E are 7 apart since we have to move 7 units of measure to get from the location at 0 to that at +7.

Also, we can construct line segment CE and note that it has a length of 7.

So, two points which are 7 apart can be connected by a line segment of length 7.

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A B C D E F

Distance and Length

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Any line segment which has a length of 7 will be congruent with CE, even if it needs to be rotated or moved to overlap it.

All such segments have the same length regardless of orientation.

So, segment CE and EC are congruent and have length 7.

Distance and Length

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A B C D E F

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What is the distance of the line below?

Is that answer positive or negative?

Distance and Length

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A B C D E F

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All measures of distance and length are positive, regardless of the direction and orientation of the points with respect to one another or

that of a line segment.

Two points cannot be a negative distance apart.

Nor can a line segment have a negative length.

Distance and Length

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A B C D E F

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You can imagine that each number on the number line is a step, and the distance between any two points is just how many steps you need

to take to get from one to the other.

Which direction you walk along the line doesn't change the distance.

Distance is always a positive number.

Do you remember a term we use in physics to describe a distance which has a direction and could have a negative value?

Distance

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A B C D E F

Ans

wer

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48 What is the location of point F?

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A B C D E F

Ans

wer

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49 What is the location of point A?

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A B C D E F

Ans

wer

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50 What is the distance from A to C?

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A B C D E F

Ans

wer

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51 What is the distance from B to E?

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A B C D E F

Ans

wer

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52 What is the distance from B to A?

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A B C D E F

Ans

wer

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Calculating Distance

Sometimes it is easier to calculate the distance between two points rather than count the steps between them.

· First, subtract the locations of the two points

· Then, take the absolute value of your answer, so that it is positive.

Remember, distance is always positive.

If you drive 100 miles, you use the same amount of energy regardless of which direction you drive...only how far you drive matters.

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Calculating DistanceLet's calculate the distance between A and C.

· First, note that A is at -7 and C is at 0

· Then, subtract those numbers: -7 - (0) = -7

[Always put the number being subtracted in parentheses to make sure to get its sign right.]

· Then take the absolute value: the absolute value of -7 is 7.

So the distance between A and C is 7.

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A B C D E F

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Calculating DistanceLet's do the same calculation, but this time let's reverse how we do the subtraction, let's subtract A from C.

· First, let's note that A is at -7 and C is at 0

· Then, let's subtract those numbers: 0 - (-7) = +7

· Then take the absolute value: the absolute value of +7 is 7.

So the distance between A and C is 7, calculated either way.

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A B C D E F

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53 What's the distance between A and F?

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A B C D E F

Ans

wer

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54 What's the distance between two points if one is located at +125 and the other is located at -350?

Ans

wer

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55 What's the distance between two points if one is located at -540 and the other is located at -180?

Ans

wer

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cm

C EA B D

F

Find the measure of each segment in centimeters.

a.

b.

Example

= 8 - 2 = 6 cm

= 1.5 cm

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56 Find a segment that is 4 cm long.

ABCD

cm

C EA B D

F

Ans

wer

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57 Find a segment that is 6.5 cm long.

ABCD

cm

C EA B D

F

Ans

wer

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ABCD

cm

C EA B D

F

Ans

wer

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59 Find a segment that is 2 cm long.AB

C

D

cm

C EA B D

F

Ans

wer

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ABC

D

cm

C EA B D

F

Ans

wer

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61 If point F was placed at 3.5 cm on the ruler, how far from point E would it be?

A 5 cm

B 4 cm

C 3.5 cm

D 4.5 cm

cm

C EA B D

F

Ans

wer

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AB BC

AC

Segment Addition Postulate

If three points are on the same line, then one of them must be between the other two.

The two shorter segments add to the larger, as shown below.

CA B

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AB BC

AC

Adding Line Segments

If B is between A and C, then AB + BC = AC.

Alternatively

If AB + BC = AC, then B is between A and C.

CA B

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CA B D E

AB BC CD DE AE++ + =

Adding Line Segments

This works for any number of segments on a line.

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Example

CA B D E

AB CD=

BC = 6DE = 5

AE = 27Given:

BE

CD

Find:

Ans

wer

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MK= 14x - 56PM= 2x + 4

P lies between K and M on a line.

ExampleLabel the line and find x given that:

PK = x + 17

Ans

wer

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Example

P, B, L, and M are collinear and are in the following order:

a) P is between B and M

b) L is between M and P

Draw a diagram and solve for x, given:

ML = 3x +16

PL = 2x +11

BM = 3x +140

PB = 3x + 13

Ans

wer

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62 What is the length of Segment AB?

Hint: always start these problems by placing the information you have into the diagram.

CA B D E

Ans

wer

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63 What is the length of Segment DE?

CA B D E

Ans

wer

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64 What is the length of Segment CA?

CA B D E

Ans

wer

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65 What is the length of Segment CE?

CA B D EA

nsw

er

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66 What is the length of Segment CE?

CA B D E

Ans

wer

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67 What is the length of Segment DA?

CA B D E

Ans

wer

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68 What is the length of Segment BE?

CA B D E

Ans

wer

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69 X, B, and Y are collinear points, with Y between B and X. Place the points on the line and solve for x, given:

BX = 6x + 151

XY = 15x - 7

BY = x - 12

YXB

Ans

wer

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70 Q, X, and R are collinear points, with X between R and Q. Draw a diagram and solve for x, given:

XQ = 15x + 10 RQ = 2x + 131

XR = 7x +1

QXR

Ans

wer

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71 B, K, and V are collinear points, with K between V and B. Draw a diagram and solve for x, given:

KB = 5x BV = 15x + 125

KV = 4x +149

VKB

Ans

wer

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Constructions

and

LociReturn to Table of Contents

Slide 186 / 206

page0svg

Introduction to Locus

In mathematics, a locus is defined to be the set of points which satisfy a given condition.

Very often, we will set up a condition and solve for the locus of points which meet that condition.

That can be done algebraically, but it can also be done with the use of drawing equipment such as a straight edge and compass.

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The Circle as a LocusOne important example of a locus is that the set of points which is equidistant from any one point is a circle.

The point from which they are equidistant is the center of the circle.

The distance from the center, is the radius, r, of the circle.

We will learn much more about circles later, but we need to learn a bit now so we can proceed with constructions.

r

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Euclid and Circles

Third Postulate: To describe a circle with any center and distance.

This postulate says that we can draw a circle of any radius, placing

its center where we choose.

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Euclid and Circles

Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.

The straight lines referenced here are the radii which are of equal length from the center to the points on the circle

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Euclid and Circles

Definition 16: And the point is called the center of the circle.

This says that the point that is equidistant from all of the points on a circle is the center of the circle.

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Introduction to Constructions

In addition to a pencil, we will be using two tools to construct geometric figures a straight edge and a compass.

A straight edge allows us to draw a straight line, which we are allowed to do between any two points.

A compass allows us to draw a circle. Try the compass to the right.You can use the pencil to rotate the compass

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Introduction to Constructions

center

r

circle

The sharp point of a compass is placed at the center of the circle. The pencil then draws the circle.

For constructions, we will just draw a small part of a circle, an arc. We do this to take advantage of the fact that every point on that arc is equidistant from the center.We can draw multiple arcs, if needed.

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Try this!

1) Create a circle using the segment below.

F

E

M Teac

her N

otes

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H

GM

Try this!

2) Create a circle using the segment below.

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Constructing Congruent Segments

Let's use these tools to create a line segment CD which is congruent with the given line segment AB.

We will first do this with a straight edge and compass.

BA

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Constructing Congruent SegmentsFirst, use your straight edge to draw a line which is longer than AB and includes Point C, such as Line a below.

BA

a

C

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Then, stretch your compass between points A and B.

BA

a

C

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Constructing Congruent SegmentsThe compass can now be used to draw an arc with any center with the radius of AB, how do you think we could use that to create a congruent segment on Line a with C as an endpoint?

BA

a

C

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Constructing Congruent SegmentsThen, keeping the compass unchanged, place its point at C and make an arc through line a. All the points on that arc are a distance AB from C. The point where the arc intersects the line, is that distance from C and on the line.

a

C

BA

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Constructing Congruent SegmentsThen, draw Point D at the intersection of the arc and line a. Point D is on the line at a distance of AB from C.

a

C

BA

D

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Segment CD is congruent with segment AB, which was our objective.

a

C

D

BA

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Try this!

3)Construct a congruent segment on the given line.

L

M

N

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I

JK

Try this!

4) Construct a congruent segment on the given line.

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Click on the image below to watch a video demonstrating constructing congruent

segments using Dynamic Geometric Software

Dynamic Geometric Software

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Slide 206 / 206

http://njc.tl/124

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