Slide 1 / 206 Slide 2 / 206 Geometry Points, Lines, Planes & Angles Part 1 www.njctl.org 2014-09-05 Slide 3 / 206 Table of Contents Introduction to Geometry click on the topic to go to that section Points and Lines Planes Congruence, Distance and Length Constructions and Loci Part 1 Part 2 Angles Congruent Angles Angles & Angle Addition Postulate Protractors Special Angle Pairs Proofs Special Angles Angle Bisectors & Constructions Locus & Angle Constructions Angle Bisectors Slide 4 / 206 Introduction to Geometry Return to Table of Contents Slide 5 / 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. Slide 6 / 206 The Origin of Geometry But 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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Slide 1 / 206 Slide 2 / 206
Geometry
Points, Lines, Planes & Angles
Part 1
www.njctl.org
2014-09-05
Slide 3 / 206
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
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?
Slide 8 / 206
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
Slide 9 / 206
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.
Slide 10 / 206
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
Slide 11 / 206
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.
Slide 12 / 206
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.
Slide 13 / 206
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
Slide 14 / 206
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 Geometry
Slide 14 (Answer) / 206
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 Geometry
[This object is a pull tab]
Teac
her N
otes
They developed what was needed for surveying, commerce and architecture, including building the pyramids.
Slide 15 / 206
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.
Slide 16 / 206
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.
Slide 17 / 206
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?
Slide 18 / 206
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."
Slide 19 / 206
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.
Slide 20 / 206
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.
Slide 21 / 206
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.
Slide 22 / 206
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.
Slide 23 / 206
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).
Slide 24 / 206
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.
Slide 25 / 206
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?
Slide 26 / 206
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.
Slide 27 / 206
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.
Slide 28 / 206
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)
Slide 29 / 206
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.
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?
Slide 43 / 206
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.
Slide 44 / 206
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.
Slide 45 / 206
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.
Slide 46 / 206
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
Slide 47 / 206
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.
A Bendpoint endpoint
AB or BA
Slide 48 / 206
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
Slide 49 / 206
A B
In this example, line Segment AB is extended in both directions to create Line AB.
Lines
A Bendpoint endpoint
Slide 50 / 206
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.
Slide 51 / 206
Give 7 different names for this line.
Example
U
W
V
b
Slide 51 (Answer) / 206
Give 7 different names for this line.
Example
U
W
V
b
[This object is a pull tab]
Ans
wer
b
Slide 52 / 206
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
Slide 52 (Answer) / 206
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
[This object is a pull tab]
Ans
wer Points D, E, and F are collinear.
Points A, B, and C are not.
Slide 53 / 206
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 Points
Slide 53 (Answer) / 206
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 Points
[This object is a pull tab]
Ans
wer
No, because a line can always be drawn between
any two points.
It's only when there are three or more points that they may not be collinear.
Slide 54 / 206
1 How many points are needed to define a line?
Slide 54 (Answer) / 206
1 How many points are needed to define a line?
[This object is a pull tab]
Ans
wer
2 points
Slide 55 / 206
2 Can there be two points which are not collinear on some line?
Yes
No
Slide 55 (Answer) / 206
2 Can there be two points which are not collinear on some line?
Yes
No
[This object is a pull tab]
Ans
wer
No
Slide 56 / 206
3 Can there be three points which are not collinear on some line?
YesNo
Slide 56 (Answer) / 206
3 Can there be three points which are not collinear on some line?
YesNo
[This object is a pull tab]
Ans
wer
No, each pair of points can make up one line, but if 3 or more points are not collinear, then they cannot be one the same line.
Slide 57 / 206
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
Slide 58 / 206
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.
Slide 59 / 206
Is it possible for two different lines to intersect at more than one point?
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.
Slide 60 / 206
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.
Slide 61 / 206
Is it possible for two different lines to intersect at more than one point?
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.
Slide 62 / 206
Is it possible for two different lines to intersect at more than one point?
Intersecting Lines
Q
T
K
R
S
So, two different lines either:
· Intersect at no points
· Intersect at one point.
F
E
D
C
Slide 63 / 206
4 What is the maximum number of points at which two distinct lines can intersect?
Slide 63 (Answer) / 206
4 What is the maximum number of points at which two distinct lines can intersect?
[This object is a pull tab]
Ans
wer
One intersection point
Slide 64 / 206
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
Slide 64 (Answer) / 206
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
[This object is a pull tab]
Ans
wer
C points A, D & C
Slide 65 / 206
6 At which point, or points, do the drawn lines intersect?
A A and DB A and CC DD none
A
CD
B
Slide 65 (Answer) / 206
6 At which point, or points, do the drawn lines intersect?
A A and DB A and CC DD none
A
CD
B
[This object is a pull tab]
Ans
wer
C point D
Slide 66 / 206
Below, the segment AB is extended to infinity, beyond Point B, to create Ray AB.
Rays
A B
A Bendpoint endpoint
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.
Slide 67 / 206
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
Slide 68 / 206
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
Slide 69 / 206
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
Slide 70 / 206
Rays AB and BA are NOT the same. What is the difference between them?
Naming Rays
A BAB
A BBA
Slide 71 / 206
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
Slide 72 / 206
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
Slide 73 / 206
7 Name a point which is collinear with points G & H.
A
BCDEFGH
C
DG
A
FH
B
E
Slide 73 (Answer) / 206
7 Name a point which is collinear with points G & H.
A
BCDEFGH
C
DG
A
FH
B
E[This object is a pull tab]
Ans
wer A
Slide 74 / 206
8 Name a point which is collinear with points D & A.
A
BCDEFGH
C
DG
A
FH
B
E
Slide 74 (Answer) / 206
8 Name a point which is collinear with points D & A.
A
BCDEFGH
C
DG
A
FH
B
E [This object is a pull tab]
Ans
wer F
Slide 75 / 206
9 Name a point which is collinear with points D & E.
A
BCDEFGH
C
DG
A
FH
B
E
Slide 75 (Answer) / 206
9 Name a point which is collinear with points D & E.
A
BCDEFGH
C
DG
A
FH
B
E [This object is a pull tab]
Ans
wer
B
Slide 76 / 206
10 Name a point which is collinear with points C & G.
A
BCDEFGH
C
DG
A
FH
B
E
Slide 76 (Answer) / 206
10 Name a point which is collinear with points C & G.
A
BCDEFGH
C
DG
A
FH
B
E[This object is a pull tab]
Ans
wer
E
Slide 77 / 206
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
Slide 77 (Answer) / 206
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
[This object is a pull tab]
Ans
wer
B Ray MO
Slide 78 / 206
12 Name an opposite ray to Ray PS.A Ray MQB Ray MOC Ray POD Ray PR
O
QP
M
TR
N
S
Slide 78 (Answer) / 206
12 Name an opposite ray to Ray PS.A Ray MQB Ray MOC Ray POD Ray PR
O
QP
M
TR
N
S
[This object is a pull tab]
Ans
wer
C Ray PO
Slide 79 / 206
13 Name an opposite ray to Ray PM.A Ray MQB Ray MOC Ray POD Ray PR
O
QP
M
TR
N
S
Slide 79 (Answer) / 206
13 Name an opposite ray to Ray PM.A Ray MQB Ray MOC Ray POD Ray PR
O
QP
M
TR
N
S
[This object is a pull tab]
Ans
wer
D Ray PR
Slide 80 / 206
14 Rays HE and HF are the same. True
False
D
H
g
P
G
E
F
p
Slide 80 (Answer) / 206
14 Rays HE and HF are the same. True
False
D
H
g
P
G
E
F
p
[This object is a pull tab]
Ans
wer
False
Slide 81 / 206
15 Rays HE and HP are the same. True
False
D
H
g
P
G
E
F
p
Slide 81 (Answer) / 206
15 Rays HE and HP are the same. True
False
D
H
g
P
G
E
F
p
[This object is a pull tab]
Ans
wer
True
Slide 82 / 206
16 Lines EH and EF are the same.True
False
D
H
g
P
G
E
F
p
Slide 82 (Answer) / 206
16 Lines EH and EF are the same.True
False
D
H
g
P
G
E
F
p
[This object is a pull tab]
Ans
wer
True
Slide 83 / 206
17 Line p contains just three points.
True
False
D
H
g
P
G
E
F
p
Slide 83 (Answer) / 206
17 Line p contains just three points.
True
False
D
H
g
P
G
E
F
p
[This object is a pull tab]
Ans
wer
False
Slide 84 / 206
18 Points D, H, and E are collinear.
True
False
D
H
g
P
G
E
F
p
Slide 84 (Answer) / 206
18 Points D, H, and E are collinear.
True
False
D
H
g
P
G
E
F
p
[This object is a pull tab]
Ans
wer
False
Slide 85 / 206
19 Points G, D, and H are collinear.
True
False
D
H
g
P
G
E
F
p
Slide 85 (Answer) / 206
19 Points G, D, and H are collinear.
True
False
D
H
g
P
G
E
F
p
[This object is a pull tab]
Ans
wer
True
Slide 86 / 206
20 Are ray LJ and ray JL opposite rays?
Yes
No
J
K
L
Slide 86 (Answer) / 206
20 Are ray LJ and ray JL opposite rays?
Yes
No
J
K
L
[This object is a pull tab]
Ans
wer No, Opposite Rays have
same endpoint but point in opposite directions
Slide 87 / 206
21 Which of the following are opposite rays?
A JK & LK
B JK & LK
C KJ & KL
D JL & KL
J
K
L
Slide 87 (Answer) / 206
21 Which of the following are opposite rays?
A JK & LK
B JK & LK
C KJ & KL
D JL & KL
J
K
L
[This object is a pull tab]
Ans
wer
C
Slide 88 / 206
22 Name the initial point of ray AC.
A
B
C
A
B
C
Slide 88 (Answer) / 206
22 Name the initial point of ray AC.
A
B
C
A
B
C[This object is a pull tab]
Ans
wer
A
Slide 89 / 206
23 Name the initial point of ray BC.
A
B
C
A
B
C
Slide 89 (Answer) / 206
23 Name the initial point of ray BC.
A
B
C
A
B
C[This object is a pull tab]
Ans
wer
B
Slide 90 / 206
Planes
Return to Table of Contents
Slide 91 / 206
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.
Slide 92 / 206
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?
Slide 93 / 206
Planes
Just as the ends of lines are points, the edges of planes are lines.
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.
Slide 95 / 206
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
Slide 96 / 206
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."
Slide 97 / 206
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.
Slide 98 / 206
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
Slide 99 / 206
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
Slide 99 (Answer) / 206
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
[This object is a pull tab]
Ans
wer
No, because a plane can always be drawn which
contains any three points.
It's only when there are four or more points that they may
not be coplanar.
Slide 100 / 206
24 How many points are needed to define a plane?
Slide 100 (Answer) / 206
24 How many points are needed to define a plane?
[This object is a pull tab]
Ans
wer
3 points
Slide 101 / 206
25 Can there be three points which are not coplanar on any plane?
YesNo
Slide 101 (Answer) / 206
25 Can there be three points which are not coplanar on any plane?
YesNo
[This object is a pull tab]
Ans
wer
No
Slide 102 / 206
26 Can there be four points which are not coplaner on any plane?
YesNo
Slide 102 (Answer) / 206
26 Can there be four points which are not coplaner on any plane?
YesNo
[This object is a pull tab]
Ans
wer
No, if the 4 points are not coplanar, then one of the points is not in the same plane as the other 3.
Slide 103 / 206
What would the intersection of two planes look like?
Hint: the walls and ceiling of this room could represent planes.
Intersecting Planes
Slide 103 (Answer) / 206
What would the intersection of two planes look like?
Hint: the walls and ceiling of this room could represent planes.
Intersecting Planes
[This object is a pull tab]
Ans
wer The intersection
of any two planes is a line.
Slide 104 / 206
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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Various Planes Defined by 3 points
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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Various Planes Defined by 3 points
Plane AZW
What are the 3 other ways you can name this
same plane?
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Various Planes Defined by 3 points Draw Plane UYA in the below drawing.
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Various Planes Defined by 3 points
Plane UYA
What are the 3 other ways you can name this
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
What are the 3 other ways you can name this
same plane?
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27 Name the point that is not in plane ABC.
A
BCD
A
B
C
D
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27 Name the point that is not in plane ABC.
A
BCD
A
B
C
D
[This object is a pull tab]
Ans
wer
D
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28 Name the point that is not in plane DBC.
A
BCD
A
B
C
D
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28 Name the point that is not in plane DBC.
A
BCD
A
B
C
D
[This object is a pull tab]
Ans
wer
A
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29 Name two points that are in both indicated planes.ABCD
A
B
C
D
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29 Name two points that are in both indicated planes.ABCD
A
B
C
D
[This object is a pull tab]
Ans
wer
B and C
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30 Name two points that are not on Line BC.ABCD
A
B
C
D
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30 Name two points that are not on Line BC.ABCD
A
B
C
D
[This object is a pull tab]
Ans
wer
A and D
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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
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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
[This object is a pull tab]
Ans
wer
No
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32 Plane LMN does not contain point P. Are points P, M, and N coplanar?
Yes
No
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32 Plane LMN does not contain point P. Are points P, M, and N coplanar?
Yes
No
[This object is a pull tab]
Ans
wer Yes on
Plane MNP
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33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture)
Yes
No
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33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture)
Yes
No
[This object is a pull tab]
Ans
wer
Yes
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34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar?
Yes
No
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34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar?
Yes
No
[This object is a pull tab]
Ans
wer
No
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35 Line BA and line DB intersect at Point ____.
ABCDEFGH
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35 Line BA and line DB intersect at Point ____.
ABCDEFGH
[This object is a pull tab]
Ans
wer
B
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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
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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
[This object is a pull tab]
Ans
wer
C
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37 Are lines EF and CD coplanar on the cube below?
Yes
No
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37 Are lines EF and CD coplanar on the cube below?
Yes
No
[This object is a pull tab]
Yes
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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
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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
[This object is a pull tab]
Ans
wer
B
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39 Planes ABC, GCD, and EGC intersect at _____?
A line GC B point C
C point A
D line AC
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39 Planes ABC, GCD, and EGC intersect at _____?
A line GC B point C
C point A
D line AC
[This object is a pull tab]
Ans
wer
B
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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
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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
[This object is a pull tab]
Ans
wer
Slide 127 / 206
41 Name a point that is coplanar with points E, F, and C.
A
B
C
D
E
F
G
H
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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
[This object is a pull tab]
Ans
wer
C
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42 Intersecting lines are __________ coplanar.
A Always
B Sometimes
C Never
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42 Intersecting lines are __________ coplanar.
A Always
B Sometimes
C Never
[This object is a pull tab]
Ans
wer
A
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43 Two planes ____________ intersect at exactly one point.
A Always
B Sometimes
C Never
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43 Two planes ____________ intersect at exactly one point.
A Always
B Sometimes
C Never
[This object is a pull tab]
Ans
wer
C
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44 A plane can __________ be drawn so that any three points are coplaner
A Always
B Sometimes
C Never
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44 A plane can __________ be drawn so that any three points are coplaner
A Always
B Sometimes
C Never
[This object is a pull tab]
Ans
wer
A
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45 A plane containing two points of a line __________ contains the entire line.
A Always
B Sometimes
C Never
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45 A plane containing two points of a line __________ contains the entire line.
A Always
B Sometimes
C Never
[This object is a pull tab]
Ans
wer
A
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46 Four points are ____________ noncoplanar.
A Always
B Sometimes
C Never
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46 Four points are ____________ noncoplanar.
A Always
B Sometimes
C Never
[This object is a pull tab]
Ans
wer
B
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47 Two lines ________________ meet at more than one point.
A Always
B Sometimes
C Never
Slide 133 (Answer) / 206
47 Two lines ________________ meet at more than one point.
A Always
B Sometimes
C Never
[This object is a pull tab]
Ans
wer
B
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Congruence, Distance and Length
Return to Table of Contents
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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
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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.
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
Slide 194 (Answer) / 206
Try this!
1) Create a circle using the segment below.
F
E
M
[This object is a pull tab]
Teac
her N
otes
The file for the "Try This!" problems is located on the
NJCTL website:https://njctl.org/courses/math/
geometry/points-lines-and-planes/
Called "Constructions Worksheet" in the "Handouts"
section.
Slide 195 / 206
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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Constructing Congruent Segments
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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Constructing Congruent Segments
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
Slide 204 / 206
I
JK
Try this!
4) Construct a congruent segment on the given line.
Slide 205 / 206
Click on the image below to watch a video demonstrating constructing congruent