SLIDE 1 Slide 1 / 209
Geometry
Points, Lines & Planes
2015-10-21 www.njctl.org
Slide 2 / 209 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 PARCC Sample Questions
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SLIDE 2 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of
MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of
MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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Math Practice
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Introduction to Geometry
Return to Table
Slide 5 / 209
SLIDE 3 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 / 209 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.
Slide 7 / 209 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
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SLIDE 4
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 / 209
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 Slide 11 / 209
SLIDE 5 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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Math Practice
This Lab addresses MP4 & MP5
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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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SLIDE 6 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
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 Geometry Slide 15 / 209
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
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Teacher Notes
They developed what was needed for surveying, commerce and architecture, including building the pyramids.
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SLIDE 7
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 / 209 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 / 209 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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SLIDE 8
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 / 209 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 / 209 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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SLIDE 9
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 / 209 Euclidean Geometry
About 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 / 209 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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SLIDE 10
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 / 209 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 / 209 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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SLIDE 11 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) Slide 29 / 209
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
- bviously 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.
Euclid's Axioms (Common Understandings) Slide 30 / 209
SLIDE 12 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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SLIDE 13
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.
Euclid's Axioms (Common Understandings) Slide 36 / 209
SLIDE 14 Points and Lines
Return to Table
Slide 37 / 209 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.
Slide 38 / 209 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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SLIDE 15 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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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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Math Practice
Question on this slide addresses MP3
Slide 40 (Answer) / 209 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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SLIDE 16 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.
Slide 42 / 209 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?
Slide 43 / 209 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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Math Practice
Question on this slide addresses MP6 & MP7
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SLIDE 17 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
- r 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.
Slide 45 / 209 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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SLIDE 18 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 B endpoint endpoint
Slide 47 / 209 Naming Line Segments
For instance, AB and BA are different names for the same segment. A line segment is named by its two endpoints. The order of the endpoints doesn't matter.
A B endpoint endpoint AB or BA
Slide 48 / 209 Naming Line Segments
For instance, AB and BA are different names for the same segment. A line segment is named by its two endpoints. The order of the endpoints doesn't matter.
A B endpoint endpoint AB or BA
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Math Practice
MP6 Remind students throughout this lesson about the proper notation and letter order (if required) for naming segments, rays & lines.
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SLIDE 19
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
A B endpoint endpoint
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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 FE ED 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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SLIDE 20 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 FE ED 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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Math Practice
MP6 Remind students throughout this lesson about the proper notation and letter order (if required) for naming segments, rays & lines.
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Give 7 different names for this line.
Example
U W V b
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Give 7 different names for this line.
Example
U W V b
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Answer b
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SLIDE 21 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
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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
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Answer
Points D, E, and F are collinear. Points A, B, and C are not.
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Is it possible for any two points to not be collinear on at least
Come up with an answer at your table. Remember, only use facts to make your argument!
Collinear Points Slide 54 / 209
SLIDE 22 Is it possible for any two points to not be collinear on at least
Come up with an answer at your table. Remember, only use facts to make your argument!
Collinear Points
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Answer
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.
Questions/comments on this slide address MP3 & MP7
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1 How many points are needed to define a line?
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1 How many points are needed to define a line?
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Answer
2 points Slide 55 (Answer) / 209
SLIDE 23 2 Can there be two points which are not collinear on some line? Yes No
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2 Can there be two points which are not collinear on some line? Yes No
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Answer
No Slide 56 (Answer) / 209
3 Can there be three points which are not collinear on some line? Yes No
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SLIDE 24 3 Can there be three points which are not collinear on some line? Yes No
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Answer
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.
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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
Reductio ad absurdum
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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
Reductio ad absurdum
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Math Practice
These next 5 slides, including the current
- ne address MP1, MP2 & MP3.
Slide 58 (Answer) / 209
SLIDE 25 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 follows from that assumption. If the conclusion is absurd, the assumption was false, and disproven.
Intersecting Lines Slide 59 / 209
Is it possible for two different lines to intersect at more than one point? 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 w e 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.
Intersecting Lines Slide 60 / 209 Intersecting Lines
We could then extend our Segment AB infinitely in both directions and our new Line AB would overlap our
- riginal 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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SLIDE 26 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.
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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
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4 What is the maximum number of points at which two distinct lines can intersect?
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SLIDE 27 4 What is the maximum number of points at which two distinct lines can intersect?
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Answer
One intersection point
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5 Which sets of points are collinear on the lines drawn in this diagram? A C D B A A, D, B B C, D, B C A, D, C D none
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5 Which sets of points are collinear on the lines drawn in this diagram? A C D B A A, D, B B C, D, B C A, D, C D none
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Answer
C Slide 65 (Answer) / 209
SLIDE 28 6 At which point, or points, do the drawn lines intersect? A A and D B A and C C none D D A C D B
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6 At which point, or points, do the drawn lines intersect? A A and D B A and C C none D D A C D B
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Answer
D Slide 66 (Answer) / 209
Below, the segment AB is extended to infinity, beyond Point B, to create Ray AB.
Rays
A B A B endpoint 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.
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SLIDE 29 Below, the segment AB is extended to infinity, beyond Point B, to create Ray AB.
Rays
A B A B endpoint 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.
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Math Practice
On the next 4 slides, this one included: MP6 is addressed Remind students throughout this lesson about the proper notation and letter order (if required) for naming segments, rays & lines
Slide 67 (Answer) / 209 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 / 209 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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SLIDE 30 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 B AB A B A B BA
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Rays AB and BA are NOT the same. What is the difference between them?
Naming Rays
A B AB A B BA
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Rays AB and BA are NOT the same. What is the difference between them?
Naming Rays
A B AB A B BA
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Math Practice
Question on this slide addresses MP3
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SLIDE 31 A B C
Below, suppose point C is between points A and B. Rays CA and CB are
Opposite rays are defined as being two rays with a common endpoint that point in opposite directions and form a straight line.
Opposite Rays Slide 72 / 209
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 B C
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7 Name a point which is collinear with points G & H. A B C D E F G H C D G A F H B E
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SLIDE 32 7 Name a point which is collinear with points G & H. A B C D E F G H C D G A F H B E
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Answer
A Slide 74 (Answer) / 209
8 Name a point which is collinear with points D & A. A B C D E F G H C D G A F H B E
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8 Name a point which is collinear with points D & A. A B C D E F G H C D G A F H B E
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Answer
F Slide 75 (Answer) / 209
SLIDE 33 9 Name a point which is collinear with points D & E. A B C D E F G H C D G A F H B E
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9 Name a point which is collinear with points D & E. A B C D E F G H C D G A F H B E
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Answer
B Slide 76 (Answer) / 209
10 Name a point which is collinear with points C & G. A B C D E F G H C D G A F H B E
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SLIDE 34 10 Name a point which is collinear with points C & G. A B C D E F G H C D G A F H B E
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Answer
E Slide 77 (Answer) / 209
11 Name an opposite ray to Ray MN. A Ray MQ B Ray MO C Ray RO D Ray PR O Q P M T R N S
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11 Name an opposite ray to Ray MN. A Ray MQ B Ray MO C Ray RO D Ray PR O Q P M T R N S
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Answer
B Slide 78 (Answer) / 209
SLIDE 35 12 Name an opposite ray to Ray PS. A Ray MQ B Ray MO C Ray PO D Ray PR O Q P M T R N S
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12 Name an opposite ray to Ray PS. A Ray MQ B Ray MO C Ray PO D Ray PR O Q P M T R N S
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Answer
C Slide 79 (Answer) / 209
13 Name an opposite ray to Ray PM. A Ray MQ B Ray MO C Ray PO D Ray PR O Q P M T R N S
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SLIDE 36 13 Name an opposite ray to Ray PM. A Ray MQ B Ray MO C Ray PO D Ray PR O Q P M T R N S
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Answer
D Slide 80 (Answer) / 209
14 Rays HE and HF are the same. True False D H g P G E F p
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14 Rays HE and HF are the same. True False D H g P G E F p
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Answer
False Slide 81 (Answer) / 209
SLIDE 37 15 Rays HE and HP are the same. True False D H g P G E F p
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15 Rays HE and HP are the same. True False D H g P G E F p
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Answer
True Slide 82 (Answer) / 209
16 Lines EH and EF are the same. True False D H g P G E F p
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SLIDE 38 16 Lines EH and EF are the same. True False D H g P G E F p
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Answer
True Slide 83 (Answer) / 209
17 Line p contains just three points. True False D H g P G E F p
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17 Line p contains just three points. True False D H g P G E F p
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Answer
False Slide 84 (Answer) / 209
SLIDE 39 18 Points D, H, and E are collinear. True False D H g P G E F p
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18 Points D, H, and E are collinear. True False D H g P G E F p
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Answer
False Slide 85 (Answer) / 209
19 Points G, D, and H are collinear. True False D H g P G E F p
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SLIDE 40 19 Points G, D, and H are collinear. True False D H g P G E F p
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Answer
True Slide 86 (Answer) / 209
20 Are ray LJ and ray JL opposite rays? Yes No J K L
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20 Are ray LJ and ray JL opposite rays? Yes No J K L
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Answer
No, opposite rays have same endpoint but point in opposite directions.
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SLIDE 41 21 Which of the following are opposite rays? A ray JK & ray LK B ray JK & ray LK C ray KJ & ray KL D ray JL & ray KL J K L
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21 Which of the following are opposite rays? A ray JK & ray LK B ray JK & ray LK C ray KJ & ray KL D ray JL & ray KL J K L
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Answer
C Slide 88 (Answer) / 209
22 Name the initial point of ray AC. A B C A B C
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SLIDE 42 22 Name the initial point of ray AC. A B C A B C
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Answer
A Slide 89 (Answer) / 209
23 Name the initial point of ray BC. A B C A B C
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23 Name the initial point of ray BC. A B C A B C
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Answer
B Slide 90 (Answer) / 209
SLIDE 43 Planes
Return to Table
Slide 91 / 209 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 / 209 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 / 209
SLIDE 44 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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Math Practice
Question on this slide addresses MP7. Additional questioning, if needed: What does collinear mean? (MP6) How is collinear related to coplanar? (MP7)
Slide 93 (Answer) / 209 Planes
Just as the ends of lines are points, the edges of planes are lines. Definition 6: The edges of a surface are lines.
Slide 94 / 209 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.
Slide 95 / 209
SLIDE 45 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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Math Practice
Question on this slide addresses MP7. Additional questioning, if needed: What does point mean? (MP6) What does line mean? (MP6) How are points and lines related to planes? (MP7)
Slide 95 (Answer) / 209
As you figured out earlier, coplanar points are points which fall
Coplanar Points and Lines
All of the lines and points shown here are coplanar.
D F E a A B C
Slide 96 / 209 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 / 209
SLIDE 46 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 / 209
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 / 209
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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Math Practice
Questions on this slide address MP2.
Slide 99 (Answer) / 209
SLIDE 47 Is it possible for any three points to not be coplanar with one another? Try and find 3 points
which are not coplanar.
Coplanar Points Slide 100 / 209
Is it possible for any three points to not be coplanar with one another? Try and find 3 points
which are not coplanar.
Coplanar Points
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Answer
No, because a plane can always be drawn which contains any three points. It's only when there are four
- r more points that they may
not be coplanar.
Questions on this slide address MP2 & MP3.
Slide 100 (Answer) / 209
24 How many points are needed to define a plane?
Slide 101 / 209
SLIDE 48 24 How many points are needed to define a plane?
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Answer
3 points Slide 101 (Answer) / 209
25 Can there be three points which are not coplanar on any plane? Yes No
Slide 102 / 209
25 Can there be three points which are not coplanar on any plane? Yes No
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Answer
No Slide 102 (Answer) / 209
SLIDE 49 26 Can there be four points which are not coplaner on any plane? Yes No
Slide 103 / 209
26 Can there be four points which are not coplaner on any plane? Yes No
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Answer
No, if the 4 points are not coplanar, then one
the same plane as the
Slide 103 (Answer) / 209
What would the intersection of two planes look like? Hint: the walls and ceiling of this room could represent planes.
Intersecting Planes Slide 104 / 209
SLIDE 50 What would the intersection of two planes look like? Hint: the walls and ceiling of this room could represent planes.
Intersecting Planes
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Answer
The intersection of any two planes is a line.
Questions on this slide address MP2 & MP4 Additional Q's that can also be used: What connections do you see between the walls and the ceiling? (MP4) Is this working or did you need a different model/example? (MP4)
Slide 104 (Answer) / 209
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
Slide 105 / 209 Various Planes Defined by 3 points
Imagine or shade in Plane BAW in the below drawing.
Slide 106 / 209
SLIDE 51 Various Planes Defined by 3 points
Plane BAW What are the 3 other ways you can name this same plane?
Slide 107 / 209 Various Planes Defined by 3 points
Plane BAW What are the 3 other ways you can name this same plane?
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Answer
Plane XBA, XWA, BXW, or the reordering of any of these letter combinations
Question on this slide addresses MP7
Slide 107 (Answer) / 209 Various Planes Defined by 3 points
Imagine or shade in Plane AZW in the below drawing.
Slide 108 / 209
SLIDE 52 Various Planes Defined by 3 points
Plane AZW What are the 3 other ways you can name this same plane?
Slide 109 / 209 Various Planes Defined by 3 points
Plane AZW What are the 3 other ways you can name this same plane?
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Answer
Plane WVZ, WAZ, AWV, or the reordering of any of these letter combinations
Question on this slide addresses MP7
Slide 109 (Answer) / 209 Various Planes Defined by 3 points
Draw Plane UYA in the below drawing.
Slide 110 / 209
SLIDE 53 Various Planes Defined by 3 points
Plane UYA What are the 3 other ways you can name this same plane?
Slide 111 / 209 Various Planes Defined by 3 points
Plane UYA What are the 3 other ways you can name this same plane?
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Answer
Plane UWA, WAY, WUY, or the reordering of any of these letter combinations
Question on this slide addresses MP7
Slide 111 (Answer) / 209 Various Planes Defined by 3 points
Imagine or draw Plane ABU in the below drawing.
Slide 112 / 209
SLIDE 54 Various Planes Defined by 3 points
Plane ABU What are the 3 other ways you can name this same plane?
Slide 113 / 209 Various Planes Defined by 3 points
Plane ABU What are the 3 other ways you can name this same plane?
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Answer
Plane AVU, BUV, BAV,
any of these letter combinations
Question on this slide addresses MP7
Slide 113 (Answer) / 209
27 Name the point that is not in plane ABC. A B C D A B C D
Slide 114 / 209
SLIDE 55 27 Name the point that is not in plane ABC. A B C D A B C D
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Answer
D Slide 114 (Answer) / 209
28 Name the point that is not in plane DBC. A B C D A B C D
Slide 115 / 209
28 Name the point that is not in plane DBC. A B C D A B C D
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Answer
A Slide 115 (Answer) / 209
SLIDE 56 29 Name two points that are in both indicated planes. A B C D A B C D
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29 Name two points that are in both indicated planes. A B C D A B C D
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Answer
B & C Slide 116 (Answer) / 209
30 Name two points that are not
A B C D A B C D
Slide 117 / 209
SLIDE 57 30 Name two points that are not
A B C D A B C D
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Answer
A & D Slide 117 (Answer) / 209
31 Line BC does not contain point R. Are points R, B, and C collinear? Draw the situation if it helps. Yes No
Slide 118 / 209
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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Answer
No Slide 118 (Answer) / 209
SLIDE 58 32 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No
Slide 119 / 209
32 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No
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Answer
Yes on Plane MNP Slide 119 (Answer) / 209
33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture) Yes No
Slide 120 / 209
SLIDE 59 33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture) Yes No
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Answer
Yes Slide 120 (Answer) / 209
34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar? Yes No
Slide 121 / 209
34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar? Yes No
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Answer
No Slide 121 (Answer) / 209
SLIDE 60 35 Line BA and line DB intersect at Point ____. A B C D E F G H
Slide 122 / 209
35 Line BA and line DB intersect at Point ____. A B C D E F G H
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Answer
B Slide 122 (Answer) / 209
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
Slide 123 / 209
SLIDE 61 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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Answer
C Slide 123 (Answer) / 209
37 Are lines EF and CD coplanar on the cube below? Yes No
Slide 124 / 209
37 Are lines EF and CD coplanar on the cube below? Yes No
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Yes Slide 124 (Answer) / 209
SLIDE 62 38 Plane ABC and plane DCG intersect at _____? A C B line DC C Line CG D they don't intersect
Slide 125 / 209
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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Answer
B Slide 125 (Answer) / 209
39 Planes ABC, GCD, and EGC intersect at _____? A line GC B point A C point C D line AC
Slide 126 / 209
SLIDE 63 39 Planes ABC, GCD, and EGC intersect at _____? A line GC B point A C point C D line AC
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Answer
C Slide 126 (Answer) / 209
40 Name another point that is in the same plane as points E, G, and H. A B C D E F G H
Slide 127 / 209
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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Answer
F Slide 127 (Answer) / 209
SLIDE 64 41 Name a point that is coplanar with points E, F, and C. A B C D E F G H
Slide 128 / 209
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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Answer
D Slide 128 (Answer) / 209
42 Intersecting lines are __________ coplanar. A Always B Sometimes C Never
Slide 129 / 209
SLIDE 65 42 Intersecting lines are __________ coplanar. A Always B Sometimes C Never
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Answer
A Slide 129 (Answer) / 209
43 Two planes ____________ intersect at exactly one point. A Always B Sometimes C Never
Slide 130 / 209
43 Two planes ____________ intersect at exactly one point. A Always B Sometimes C Never
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Answer
C Slide 130 (Answer) / 209
SLIDE 66 44 A plane can __________ be drawn so that any three points are coplaner. A Always B Sometimes C Never
Slide 131 / 209
44 A plane can __________ be drawn so that any three points are coplaner. A Always B Sometimes C Never
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Answer
A Slide 131 (Answer) / 209
45 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes C Never
Slide 132 / 209
SLIDE 67 45 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes C Never
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Answer
A Slide 132 (Answer) / 209
46 Four points are ____________ noncoplanar. A Always B Sometimes C Never
Slide 133 / 209
46 Four points are ____________ noncoplanar. A Always B Sometimes C Never
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Answer
B Slide 133 (Answer) / 209
SLIDE 68 47 Two lines ________________ meet at more than one point. A Always B Sometimes C Never
Slide 134 / 209
47 Two lines ________________ meet at more than one point. A Always B Sometimes C Never
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Answer
B Slide 134 (Answer) / 209
Congruence, Distance and Length
Return to Table
Slide 135 / 209
SLIDE 69
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 136 / 209
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 Slide 137 / 209
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
Slide 138 / 209
SLIDE 70 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
Slide 139 / 209
Would the same be true for any two rays?
Congruence
a b
Slide 140 / 209
Would the same be true for any two rays?
Congruence
a b
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Math Practice
Questions on this slide address MP3.
Slide 140 (Answer) / 209
SLIDE 71 Again, all rays are infinitely long, so they have the same length. And once their vertices and any other point on both rays
- verlap, all of their points will overlap.
All rays are congruent.
Congruence
a b
Slide 141 / 209
Would the same be true of all line segments?
Congruence
a b
Slide 142 / 209
Would the same be true of all line segments?
Congruence
a b
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Math Practice
Questions on this slide address MP3.
Slide 142 (Answer) / 209
SLIDE 72 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
a b
Slide 143 / 209
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: Slide 144 / 209
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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Math Practice
Questions on this slide address MP7.
Slide 144 (Answer) / 209
SLIDE 73 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 Slide 145 / 209
1 2 3 4 5 6 7 8 9 10
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 Slide 146 / 209
1 2 3 4 5 6 7 8 9 10
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 Slide 147 / 209
SLIDE 74 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. 1 2 3 4 5 6 7 8 9 10
A B C D E F
Distance and Length Slide 148 / 209
Any line segment which has a length of 7 will be congruent with segment 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
1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 149 / 209
What is the distance of the line below? Is that answer positive or negative?
Distance and Length
1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 150 / 209
SLIDE 75 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
1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 151 / 209
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
1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 152 / 209
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
1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
displacement
( X )
Question addresses MP7
Slide 152 (Answer) / 209
SLIDE 76 48 What is the location of point F? 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 153 / 209
48 What is the location of point F? 1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
+10 Slide 153 (Answer) / 209
49 What is the location of point A? 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 154 / 209
SLIDE 77 49 What is the location of point A? 1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
Slide 154 (Answer) / 209
50 What is the distance from A to C? 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 155 / 209
50 What is the distance from A to C? 1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
7 units Slide 155 (Answer) / 209
SLIDE 78 51 What is the distance from B to E? 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 156 / 209
51 What is the distance from B to E? 1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
11 units Slide 156 (Answer) / 209
52 What is the distance from B to A? 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 157 / 209
SLIDE 79 52 What is the distance from B to A? 1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
3 units Slide 157 (Answer) / 209 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
- f which direction you drive...only how far you drive matters.
Slide 158 / 209 Calculating Distance
Let'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. 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 159 / 209
SLIDE 80 Calculating Distance
Let'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. 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 160 / 209
53 What's the distance between A and F? 1 2 3 4 5 6 7 8 9 10
A B C D E F
Slide 161 / 209
53 What's the distance between A and F? 1 2 3 4 5 6 7 8 9 10
A B C D E F
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Answer
17 units Slide 161 (Answer) / 209
SLIDE 81 54 What's the distance between two points if one is located at +125 and the other is located at -350?
Slide 162 / 209
54 What's the distance between two points if one is located at +125 and the other is located at -350?
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Answer
475 units Slide 162 (Answer) / 209
55 What's the distance between two points if one is located at -540 and the other is located at -180?
Slide 163 / 209
SLIDE 82 55 What's the distance between two points if one is located at -540 and the other is located at -180?
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Answer
360 units Slide 163 (Answer) / 209
cm C E A B D F Find the measure of each segment in centimeters. a. b.
Example
CE = 8 - 2 = 6 cm AB = 1.5 cm Note: When giving the measurement of segments using an equal sign, the segment bar is not used.
Slide 164 / 209
56 Find a segment that is 4 cm long. A B C D cm C E A B D F
Slide 165 / 209
SLIDE 83 56 Find a segment that is 4 cm long. A B C D cm C E A B D F
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Answer
B Slide 165 (Answer) / 209
57 Find a segment that is 6.5 cm long. A B C D cm C E A B D F
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57 Find a segment that is 6.5 cm long. A B C D cm C E A B D F
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Answer
B Slide 166 (Answer) / 209
SLIDE 84 58 Find a segment that is 3.5 cm long. A B C D cm C E A B D F
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58 Find a segment that is 3.5 cm long. A B C D cm C E A B D F
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Answer
D Slide 167 (Answer) / 209
59 Find a segment that is 2 cm long. A B C D cm C E A B D F
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SLIDE 85 59 Find a segment that is 2 cm long. A B C D cm C E A B D F
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Answer
B Slide 168 (Answer) / 209
60 Find a segment that is 5.5 cm long. A B C D cm C E A B D F
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60 Find a segment that is 5.5 cm long. A B C D cm C E A B D F
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D Slide 169 (Answer) / 209
SLIDE 86 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 E A B D F
Slide 170 / 209
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 E A B D F
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Answer
D Slide 170 (Answer) / 209
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.
C A B
Slide 171 / 209
SLIDE 87
AB BC AC C A B
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.
Slide 172 / 209
C A B D E AB + BC + CD + DE = AE
Adding Line Segments
This works for any number of segments on a line.
Slide 173 / 209 Example
C A B D E
AB CD = BC= 6 DE = 5 AE = 27 Given: BE CD Find:
Slide 174 / 209
SLIDE 88 Example
C A B D E
AB CD = BC= 6 DE = 5 AE = 27 Given: BE CD Find:
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Answer create a variable for the 2 unknown equivalent lengths, x AB = CD = x AB + BC + CD + DE = AE x + 6 + x + 5 = 27 2x + 11 = 27 2x = 16 x = 8 units = CD BE = 6 + 8 + 5 = 19 units This example addresses MP1 & MP2
Slide 174 (Answer) / 209
MK= 14x - 56 PM= 2x + 4 P lies between K and M on a line.
Example
Label the line and find the value of x given that: PK = x + 17
Slide 175 / 209
MK= 14x - 56 PM= 2x + 4 P lies between K and M on a line.
Example
Label the line and find the value of x given that: PK = x + 17
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Answer Solve for x (x + 17) + (2x + 4) = 14x - 56 3x + 21 = 14x - 56 + 56 + 56 3x + 77 = 14x
77 = 11x 7 = x This example addresses MP 1 & MP2
Slide 175 (Answer) / 209
SLIDE 89 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
Slide 176 / 209 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
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Answer 8x + 40 = 3x + 140 5x + 40 = 140 5x = 100 x = 20 This example addresses MP1 & MP2
Slide 176 (Answer) / 209
62 What is the length of Segment AB?
Hint: always start these problems by placing the information you have into the diagram.
C A B D E
Slide 177 / 209
SLIDE 90 62 What is the length of Segment AB?
Hint: always start these problems by placing the information you have into the diagram.
C A B D E
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Answer
3 Slide 177 (Answer) / 209
63 What is the length of Segment DE? C A B D E
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63 What is the length of Segment DE? C A B D E
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11 Slide 178 (Answer) / 209
SLIDE 91 64 What is the length of Segment CA? C A B D E
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64 What is the length of Segment CA? C A B D E
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6 Slide 179 (Answer) / 209
65 What is the length of Segment CE? C A B D E
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SLIDE 92 65 What is the length of Segment CE? C A B D E
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14 Slide 180 (Answer) / 209
66 What is the length of Segment CE? C A B D E
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66 What is the length of Segment CE? C A B D E
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14 Slide 181 (Answer) / 209
SLIDE 93 67 What is the length of Segment DA? C A B D E
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67 What is the length of Segment DA? C A B D E
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9 Slide 182 (Answer) / 209
68 What is the length of Segment BE? C A B D E
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SLIDE 94 68 What is the length of Segment BE? C A B D E
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Answer
17 Slide 183 (Answer) / 209
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 Y X B
Slide 184 / 209
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 Y X B
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Answer x - 12 + 15x -7 = 6x + 151 16x - 19 = 6x + 151 10x = 170 x = 17
x - 12 15x - 7 6x + 151 Y B X
Slide 184 (Answer) / 209
SLIDE 95 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 Q X R
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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 Q X R
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Answer
X R Q
15x + 10
2x + 131
7x + 1 7x + 1 + 15x + 10 = 2x + 131 22x + 11 = 2x + 131 20x = 120 x = 6
Slide 185 (Answer) / 209
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 V K B
Slide 186 / 209
SLIDE 96 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 V K B
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Answer
K V B 5x 15x + 125 4x + 149
4x + 149 + 5x = 15x + 125 9x + 149 = 15x + 125 6x = 24 x = 4
Slide 186 (Answer) / 209
Constructions and Loci
Return to Table
Slide 187 / 209
Constructions and Loci
Return to Table
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Math Practice
This entire lesson w/ constructions addresses MP5
Slide 187 (Answer) / 209
SLIDE 97
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.
Slide 188 / 209 The Circle as a Locus
One 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
Slide 189 / 209 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.
Slide 190 / 209
SLIDE 98
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
Slide 191 / 209 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.
Slide 192 / 209 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
Slide 193 / 209
SLIDE 99 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.
Slide 194 / 209 Try this!
1) Create a circle using the segment below.
F E M
Slide 195 / 209 Try this!
1) Create a circle using the segment below.
F E M
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Teacher Notes
The file for the "Try This!" problems is located on the NJCTL website: https://njctl.org/courses/math/ geometry-2015-16/points-lines- and-planes/constructions- worksheet-for-presentation/ Called "Constructions Worksheet" in the "Handouts" section.
Slide 195 (Answer) / 209
SLIDE 100
H G M
Try this!
2) Create a circle using the segment below.
Slide 196 / 209 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.
B A
Slide 197 / 209 Constructing Congruent Segments
First, use your straight edge to draw a line which is longer than AB and includes Point C, such as Line a below.
B A a C
Slide 198 / 209
SLIDE 101
Constructing Congruent Segments
Then, stretch your compass between points A and B.
B A a C
Slide 199 / 209 Constructing Congruent Segments
The 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?
B A a C
Slide 200 / 209 Constructing Congruent Segments
Then, 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 B A
Slide 201 / 209
SLIDE 102 Constructing Congruent Segments
Then, 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 B A D
Slide 202 / 209 Constructing Congruent Segments
Segment CD is congruent with Segment AB, which was our
a C D B A
Slide 203 / 209 Try this!
3) Construct a congruent segment on the given line.
L M N
Slide 204 / 209
SLIDE 103 I J K
Try this!
4) Construct a congruent segment on the given line.
Slide 205 / 209
Click on the image below to watch a video demonstrating constructing congruent segments using Dynamic Geometric Software
Dynamic Geometric Software Slide 206 / 209 PARCC Sample Test Questions
The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing unit 1, you should be able to answer these questions. Good Luck! Return to Table
Slide 207 / 209
SLIDE 104 72 Points J, K, and L are distinct points, and JK = KL. Which
- f these statements must be true? Select all that apply.
A J, K, and L are coplanar B J, K, and L are collinear C K is the midpoint of JL D JK ≅ KL E The measure of ∠JKL is 90°. Question 11/11
Topic: Congruence, Distance & Length PARCC Released Question (PBA)
Slide 208 / 209
72 Points J, K, and L are distinct points, and JK = KL. Which
- f these statements must be true? Select all that apply.
A J, K, and L are coplanar B J, K, and L are collinear C K is the midpoint of JL D JK ≅ KL E The measure of ∠JKL is 90°. Question 11/11
Topic: Congruence, Distance & Length PARCC Released Question (PBA)
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Answer
A & D Slide 208 (Answer) / 209
Question 11/11
Topic: Congruence, Distance & Length PARCC Released Question (PBA)
Slide 209 / 209
SLIDE 105 Question 11/11
Topic: Congruence, Distance & Length PARCC Released Question (PBA)
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Answer
Slide 209 (Answer) / 209
