Geometry Points, Lines, & Planes -...

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August 23, 2015

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Geometry

Points, Lines,& Planes

2015-02-04

Table of Contents

Introduction to Geometry

Points and Lines

Planes

Congruence, Distance and Length

Constructions and Loci

Intro to Geometry

Geometry

of Contents

Points, Lines, & Planes

The Origin of Geometry

farmland.

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


The Origin of Geometry

But over thousands of yearsAfrican became desert.

The banks of the Nile became prime farmland.


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The land along the Nile became crowded with people.

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

Fertile soil due to annual flooding, which deposited silt from

trackof who owned which land?


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

Reestablishing land ownershipafter each annual flood requireda

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


ago, so let's do a lab to see how you would solve this problem.

Land Boundaries Lab

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


Before the annual flood ofthe Nile three plots ofland might be as shown.

The orange dots are toindicate stakes that wereplaced above the floodlevel.

The stakes would remainin the same location from

Plot 1


Before flooding, three plots ofland might be look like these.

Land Boundaries Lab

Plot 1

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

river had moved in its course.


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, thepharaoh would send outgeometers with ropesthat had beenmeasure eachland in prior years.

How did they do it?

of the paper or rulersbecause these were openfields of great size.)


Egyptian mathematics was verypractical. What practicalapplications do you think theEgyptians used mathematics for?

They did not develop abstract

Greeks, who built upon what theyhad learned from the Egyptians,Babylonians and others.

Egyptian Geometry


axioms.


Euclid's book, The Elements,summarized the results of Greek

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

It also represents a great place tolearn that type of thinking.


Euclidean Geometry dates prior to 400 BC.

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

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


This statement wasposted above Plato's

Athens, about 2500years ago.

This renaissancepainting by Raphaeldepicts that

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


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

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

These works were not purposelydestroyed, but deteriorated with ageas there was no central governmentto maintain them.


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.


Much of the thinking ofmodern science andmathematics developed fromthe rediscovery of Euclid'sElements.

The thinking that underliesEuclidean Geometry has heldup very well.

Many still believe it is thebest introduction to analyticalthinking.



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

Geometry underlies much of science,engineering and

mathematics (STEM).


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 proveWhat follows from what we have previously assumed or proven

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

and those who invented geometry would have wanted.


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This also represents a path to logicalthinking, which British philosopherBertrand Russell showed is identical tomathematical thinking.

Click on the image to watch a shortvideo of Bertrand Russell's messageto the future which was filmed in 1959.

Did you hear anything thatsounded familiar?

What was it?


Euclid's assumptions are postulates definitions

develop further understanding.

Major ideas which are proven are called Theorems

Ideas that easily follow from a theorem are called Corollaries


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

we will introduce them as required.

Also, additional modern terms which you will need to knowintroduced as needed.


Euclid called his axioms "Common Understandings."

factthat he wrote them down as his assumptions reflects howcarefully he wanted to make clear his thinking.

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


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 seemsobviously true...but turns out to not always be true.

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

Jul 18-10:22 PM

Things which are equal to the same thing are also

For example:

and I know that Boband Sarah are the same height...what other conclusion can I come to?

Sarah


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Jul 18-10:22 PM

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

For example,

if you and I each have the sameamount ofwe each earn the same additionalamount, let's say $2,

then we still each have the sametotalin this case $22.

Jul 18-10:22 PM

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

This is just like the second axiom.

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

Jul 18-10:22 PM

Things which coincide with one another

For example,

if I lay two pieces of woodside by side and bothand all the points inbetween line up, I would

have equal

Jul 18-10:22 PM

The whole is greater than the part.

For example,

if an object is made up of morethan one part,

then the object has to be largerthan any of those parts.

Jul 18-10:22 PM

are also

equal.

remaindersare equal.

are equal

Points & Lines

Points and Lines

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Definitions are words or terms that have anagreed upon meaning; they cannot bederived or proven.

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

When we draw objects based on these definitions, that is just to

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


It cannot be divided into smaller parts.

It is a location in space, without dimensions.

It has no length, width or height.


Look at this dot. Why can it not be considered a point?



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.

Lines can be thought of as an infinite number of points with nobetween them.


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 withendpoints, why is is not accurate?


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with the points on itself.

In a straight line the points lie next to one another withoutor turning in any direction.

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

Jul 19-12:05 PM

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

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

located at any point in space.

Jul 19-12:05 PM

continuously in a straight line.

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

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

straight edge can be used.

Line Segments

Line Segments

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


line, which extends to infinity in both directions, can becreated by extending a line segment in both directions.

This is allowed by our definitions and postulates by imaginingconnecting each endpoint of the segment to other points that liebeyond it, in both directions.


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directions to


Points, Lines, & Planes Points, Lines, & Planes

Collinear points are points which fall on the same line.

Which of these points are collinear with the drawn line?


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

Come up with an answer at your table.

Jul 18-10:49 PM

How many points are needed to define a line?


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Jul 18-10:53 PM

Can there be two points which are not collinear

on some line?

Jul 18-10:53 PM

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

Jul 18-10:59 PM

4 Which sets of points are collinear on the lines

drawn in this diagram?

Aug 27-9:19 AM

5 Name a point which is collinear with points G

& H.

Aug 27-9:19 AM

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

Aug 27-9:19 AM

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


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Aug 27-9:19 AM

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


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

Argumentum ad absurdum

Reductio ad absurdum


Argumentum ad absurdum

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 followsfrom that assumption. If the conclusion is absurd, the assumption was

false, and disproven.



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.

e can draw a line segment between any two points.

That segment would overlap both our original lines and B, since they


and our new original two lines to infinityin both directions.

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

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



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

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.


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So, two

Intersect at no points

Intersect at one point.

Jul 18-10:59 PM

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

Jul 18-10:49 PM

10 What is the maximum number of points at which two

distinct lines can intersect?


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


Naming Rays

When naming a ray the first letter is the point where the ray begins

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?


Naming Rays

Also, instead of the double-headed arrows which are used for lines,


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Naming Rays


Naming Rays

Aug 27-10:02 AM

11 Name the initial point of ray AC.

Aug 27-10:02 AM

12 Name the initial point of ray BC.


opposite rays

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


13 Are ray LJ and ray JL opposite rays?


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14

JK & LK

JK & LK

KJ & KL

Aug 25-8:23 PM

15 Name an opposite ray to Ray MN.

Ray MQ

Ray MO

Ray RO

Ray PR


collinear points.

collinear if they lie on the same line.


16

False


17 Rays HE and HP are the same.

False


18

False


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19

False


20

False


21

False

Jul 18-11:05 PM

22 Name an opposite ray to Ray PS.

Jul 18-11:05 PM

23 Name an opposite ray to Ray PM.

Planes

of Contents


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Aug 20-2:48 PM Points, Lines, & Planes

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

It can extend infinitely in the directions of its length and breadth,

But it has no height at all.


Recall that points which fall on the same line are calledcollinear points.

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


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

Definition 6: The edges of a surface are lines.


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

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

with the straight lines on itself.


are points which fallon the same plane.

All of the lines and points


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Points, Lines, & Planes Jul 18-10:49 PM

24 How many points are needed to define a plane?

Jul 18-10:53 PM

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

Jul 18-10:53 PM

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

Points, Lines, & Planes Points, Lines, & Planes

not lie on the indicatedplane, they are coplanar

Can you imagine a plane inwhich they are coplanar?

Can you draw it on theimage? What could be aname for that plane?


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Is it possible for anythree points to not becoplanar with one

another?

on this diagramwhich are not


What would the intersection of two planes look like?

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


The intersection of

other way than a line.



What are the 3 otherways you can name this

same plane?



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same plane?




same plane?




same plane?

Aug 27-10:23 AM

27 Name the point that is not in plane ABC.

C


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Aug 27-10:23 AM

28 Name the point that is not in plane DBC.

C

Aug 27-10:25 AM

29 Name two points that are

in both indicated planes.

C

Aug 27-10:25 AM

30 Name two points that are

not on Line BC.

C


31collinear? Draw the situation if it helps.


32


33


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34

Aug 27-10:31 AM

35 Line BA and line DB intersect at Point ____.


36


37


38 Plane ABC and plane DCG intersect at _____?

line DC


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


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Aug 27-10:36 AM

40 Name another point that is in the same plane as

points E, G, and H.

Aug 27-10:36 AM

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


42 Intersecting lines are __________ coplanar.


43


44 A plane can __________ be drawn so that


45 __________ contains the entire line.


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46 Four points are ____________ noncoplanar.


47 Two lines ________________ meet at more than one point.

Congruence, Distance & Length

of Contents

Line Segments

Line Segments

By this definition, it can be seen that all lines are congruent

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

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

Line Segments


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August 23, 2015

Line Segments

Therefore, they will overlap at every point once they are rotated

Line Segments

Line Segments

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

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

All rays are congruent.

Line Segments

Line Segments

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

Only segments with the same length are congruent.

Line Segments

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

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

Distance:

Length:


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Line Segments Line Segments

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


We can say that points C and E are 7 apart since we have to move 7

Line Segments

Any line segment which has a length of 7 will be congruent withsegment 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.

Line Segments

What is the distance of the line below?

Is that answer positive or negative?


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Line Segments

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

that of a line segment.

Nor can a line segment have a negative length.

Line Segments

the distance between any two points is just how many steps you need

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

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

Jul 12-10:24 PM

48

Jul 12-10:24 PM

49

Jul 12-10:37 PM

50 What is the distance from A to C?

Jul 12-10:37 PM

51


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Jul 12-10:37 PM

52

Line Segments

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

First, subtract the locations of the two points

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

Line Segments

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

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

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

Line Segments

Jul 12-10:56 PM

53 What's the distance between A and F?

Jul 12-10:56 PM

54


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Jul 12-10:56 PM

55

Line Segments

CE = 8 - 2 = 6 cm

AB = 1.5 cm

Note: When giving the measurement of segments usingan equal sign, the segment bar is not used.

Line Segments

56

Line Segments

57 Find a segment that is 6.5 cm long.

Line Segments


Line Segments

59 Find a segment that is 2 cm long.


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Line Segments


Line Segments

61

Line Segments

BC

AC

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

Line Segments

BC

AC

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

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

Line Segments

BC

Line Segments

AB CD

BC= 6

DE = 5

AE = 27Given:

BE

CD

Find:


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Line Segments

Label the line and find the value of x given that:

Line Segments

Line Segments

62

Hint: always start these problemsby placing the information youhave into the diagram.

Line Segments

63

Line Segments

64

Line Segments

65


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Line Segments

66

Line Segments

67

Line Segments

68

Line Segments

69

BX = 6x + 151

XY = 15x - 7

Line Segments

70

XQ = 15x + 10

RQ = 2x + 131

Line Segments

71

KB = 5x

KV = 4x +149


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August 23, 2015

Constructions & Loci

Constructions

and

of Contents

Locus&

Constructions

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

which meet that condition.

ofdrawing equipment such as a straight edge and compass.

Locus&

Constructions

Locus&

Constructions

center and distance.

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

its center where we choose.

Locus&

Constructions

such that all the straight lines falling upon it from one pointamong those

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

Locus&

Constructions

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


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August 23, 2015

Locus&

Constructions

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

allowedto do between any two points.

0°

168

Locus&

Constructions

center

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

For constructions, we will just draw a small part of a circle, an arc. Wedo this to take advantage of the fact that every point on that arc is

0°

172

Aug 18-8:55 PM Aug 18-8:55 PM



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August 23, 2015

Line Segments

0°

172

Line Segments

0°

172

Line Segments

0°172

Line Segments

Line Segments

objective.

Aug 18-8:50 PM


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August 23, 2015

Aug 18-8:53 PM Video

Click on the image below to watch a videodemonstrating constructing congruent

segments using Dynamic Geometric Software

Dynamic Geometric Software

PARCC Sample Questions

The remaining slides in this presentation contain questions from the

answer these questions.

Good Luck!

of Contents

Feb 4-7:14 AM

72 Points J, K, and L are distinct points, and JK = KL. Which of these statements must be true? Select all that apply.

J, K, and L are coplanar

J, K, and L are collinear

C K is the midpoint of JL

D JK ≅ KLThe measure of ∠JKL is 90°.

Feb 2-1:03 PM Oct 1-12:47 PM

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