Geometry Points, Lines, - NJCTL
Slide 1 / 206 Slide 2 / 206 Geometry Points, Lines, Planes & Angles Part 1 2014-09-05 www.njctl.org Part 1 Table of Contents Introduction to Geometry Points and Lines Planes Congruence, Distance and Length Constructions and Loci Part 2 Angles Congruent Angles Angles & Angle Addition Postulate Protractors Special Angle Pairs Proofs Special Angles Angle Bisectors Locus & Angle Constructions Angle Bisectors & Constructions click on the topic to go to that section Slide 3 / 206
Slide 4 / 206 Introduction to Geometry Return to Table of Contents The Origin of Geometry Slide 5 / 206 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. 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 6 / 206
The Origin of Geometry Slide 7 / 206 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? Egyptian Geometry 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. Slide 9 / 206 Land Boundaries Lab 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. 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.
Land Boundaries Lab Slide 10 / 206 Pre- Flood Boundary Map 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 B Plot 2 C D Plot 3 E Land Boundaries Lab Before flooding, three plots of land might be look like these. Pre- Flood Boundary Map Slide 11 / 206 Afterwards, only the stakes above the flood level remained, and the river had moved in its course. Post-Flood Map of River and Markers A A Plot 1 B B Plot 2 C C D D Plot 3 E E Land Boundaries Lab 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. The Egyptians only had stakes and rope, you only have tape and string. Slide 12 / 206
Slide 13 / 206 Land Boundaries Lab After the flood, the pharaoh would send out geometers with ropes that had been used to measure each plot of land in prior years. Post-Flood Map of River and Markers A B How did they do it? (You can't use the edges of the paper or rulers because these were open fields of great size.) C D E Slide 14 / 206 Egyptian mathematics was very practical. What practical applications do you think the Egyptians used mathematics for? Teacher Notes Egyptian Geometry 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. 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 15 / 206
Euclidean Geometry Slide 16 / 206 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. Euclidean Geometry Slide 17 / 206 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? Euclidean Geometry "Let none who are ignorant of geometry enter here." This statement was posted above Plato's Academy, in ancient Athens, about 2500 years ago. This renaissance painting by Raphael depicts that academy. Slide 18 / 206
Euclidean Geometry Slide 19 / 206 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. Euclidean Geometry Slide 20 / 206 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. 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 21 / 206
Euclidean Geometry Slide 22 / 206 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. Euclidean Geometry Slide 23 / 206 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). 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 24 / 206
Euclidean Geometry Slide 25 / 206 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? Euclidean Geometry Slide 26 / 206 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. 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 27 / 206
Slide 28 / 206 Euclid's Axioms (Common Understandings) 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. Slide 29 / 206 Euclid's Axioms (Common Understandings) 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. Slide 30 / 206 Euclid's First Axiom Things which are equal to the same thing are also equal to one another. 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
Slide 31 / 206 Euclid's Second Axiom If equals are added to equals, the whole are equal. 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. Slide 32 / 206 Euclid's Third Axiom If equals be subtracted from equals, the remainders are equal. This is just like the second axiom. Come up with an example on your own. Look back at the second axiom if you need a hint. Slide 33 / 206 Euclid's Fourth Axiom Things which coincide with one another are equal to one another. 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.
Slide 34 / 206 Euclid's Fifth Axiom The whole is greater than the part. 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. Slide 35 / 206 Euclid's Axioms (Common Understandings) 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. Slide 36 / 206 Points and Lines Return to Table of Contents
Definitions Slide 37 / 206 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. Points Slide 38 / 206 Definition 1: A point is that which has no part. 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. 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. Slide 39 / 206
Slide 40 / 206 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 Lines Slide 41 / 206 Definition 2: A line is breadthless length. 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. Lines can be thought of as an infinite number of points with no space between them. Lines Definition 3: The ends of a line are points. 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. Even though this is how we correctly depict a line with endpoints, why is is not accurate? Slide 42 / 206
Lines Slide 43 / 206 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 Lines First Postulate: To draw a line from any point to any point. 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 B endpoint endpoint Slide 47 / 206 Naming Line Segments AB or BA A B endpoint endpoint A line segment is named by its two endpoints. The order of the endpoints doesn't matter. For instance, AB and BA are different names for the same segment. Lines 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. Slide 48 / 206
Slide 49 / 206 Lines In this example, line Segment AB is extended in both directions to create Line AB. A B endpoint endpoint B A Slide 50 / 206 Naming Lines 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. When using two points to name a line, their order doesn't matter since the line goes in both directions. F D E Here are 7 valid names for this line. DE DF EF ED FD FE a a Slide 51 / 206 Example Give 7 different names for this line. Answer b U V W
Slide 52 / 206 Collinear Points Collinear points are points which fall on the same line. B F C A D E Answer Which of these points are collinear with the drawn line? a 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! Slide 54 / 206 How many points are needed to define a line? Answer 1 Answer Collinear Points
Slide 55 / 206 Can there be two points which are not collinear on some line? Answer 2 Yes No Slide 56 / 206 Can there be three points which are not collinear on some line? Answer 3 Yes No Intersecting Lines Is it possible for two different lines to intersect at more than one point? A good technique to prove whether this is possible is called either Argumentum ad absurdum or Reductio ad absurdum Slide 57 / 206
Intersecting Lines Slide 58 / 206 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. Intersecting Lines Slide 59 / 206 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 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. 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 60 / 206
Slide 61 / 206 Intersecting Lines Is it possible for two different lines to intersect at more than one point? 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 Intersecting Lines Is it possible for two different lines to intersect at more than one point? E So, two different lines either: D F · Intersect at no points · Intersect at one point. C R Q K S What is the maximum number of points at which two distinct lines can intersect? Answer 4 T Slide 63 / 206
Slide 64 / 206 Which sets of points are collinear on the lines drawn in this diagram? A B C D A, D, B C, D, B A, D, C none Answer 5 A D C B Slide 65 / 206 6 At which point, or points, do the drawn lines intersect? A and D A and C D none Answer A B C D A D C B Slide 66 / 206 Rays 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. Below, the segment AB is extended to infinity, beyond Point B, to create Ray AB. A endpoint A B endpoint B
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? Line AB or Line BA B A Ray AB B A 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. AB or BA B A AB B A Slide 69 / 206 Naming Rays B A Segment AB can be extended in either in either direction. We can extend it at B to get ray AB. A AB B Or, we can extend it at A to get Ray BA. A BA B
Slide 70 / 206 Naming Rays A A AB BA B B Rays AB and BA are NOT the same. What is the difference between them? Opposite Rays Slide 71 / 206 Opposite rays are defined as being two rays with a common endpoint that point in opposite directions and form a straight line. Below, suppose point C is between points A and B. A C B Rays CA and CB are opposite rays. Slide 72 / 206 Collinear Rays A C B 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.
Slide 73 / 206 A G E HD F B C C G H E A B Answer 7 Name a point which is collinear with points G & H. D F Slide 74 / 206 A G E HD F B C C G H E A B Answer 8 Name a point which is collinear with points D & A. D F Slide 75 / 206 A G E HD F B C C G H E A D F B Answer 9 Name a point which is collinear with points D & E.
Slide 76 / 206 A G E HD F B C C A B D G H Answer 10 Name a point which is collinear with points C & G. E F Slide 77 / 206 11 Name an opposite ray to Ray MN. Answer A Ray MQ B Ray MO C Ray RO D Ray PR O P R S M N Q T Slide 78 / 206 12 Name an opposite ray to Ray PS. Ray MQ Ray MO Ray PO Ray PR Answer A B C D O P R S M Q T N
Slide 79 / 206 13 Name an opposite ray to Ray PM. Ray MQ Ray MO Ray PO Ray PR Answer A B C D O M R N Q P S T Slide 80 / 206 14 Rays HE and HF are the same. True Answer False p P D E H G F g Slide 81 / 206 15 Rays HE and HP are the same. True False Answer p P D E H F G g
Slide 82 / 206 16 Lines EH and EF are the same. True False Answer p P D E H G F g Slide 83 / 206 17 Line p contains just three points. True False Answer p P D E H G F g Slide 84 / 206 18 Points D, H, and E are collinear. True False Answer p P D E H F G g
Slide 85 / 206 19 Points G, D, and H are collinear. True False Answer p P D E H G F g Slide 86 / 206 20 Are ray LJ and ray JL opposite rays? Yes Answer No J K L Slide 87 / 206 21 Which of the following are opposite rays? C KJ & KL B JK & LK D JL & KL Answer A JK & LK J K L
Slide 88 / 206 22 Name the initial point of ray AC. A C Answer AB B C Slide 89 / 206 23 Name the initial point of ray BC. A C Answer AB B C Slide 90 / 206 Planes Return to Table of Contents
Planes Slide 91 / 206 Definition 5: A surface is that which has length and breadth only. 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. Planes Slide 92 / 206 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? Planes Definition 6: The edges of a surface are lines. Just as the ends of lines are points, the edges of planes are lines. Slide 93 / 206
Slide 94 / 206 Planes Definition 7: A plane surface is a surface which lies evenly with the straight lines on itself. 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? Coplanar Points and Lines Slide 95 / 206 As you figured out earlier, coplanar points are points which fall on the same plane. B C F A E D All of the lines and points shown here are coplanar. a Naming Planes Planes can be named by any three points that are not collinear. This plane can be named "Plane KMN," "Plane LKM," or "Plane KNL." Also, it can be named by the single letter, "Plane R." Slide 96 / 206
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 Coplanar Points 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? 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. Answer Coplanar Points
Slide 100 / 206 Answer 24 How many points are needed to define a plane? Slide 101 / 206 Yes No Answer 25 Can there be three points which are not coplanar on any plane? Slide 102 / 206 Yes No Answer 26 Can there be four points which are not coplaner on any plane?
Slide 103 / 206 Intersecting Planes Hint: the walls and ceiling of this room could represent planes. Answer What would the intersection of two planes look like? Slide 104 / 206 Intersecting Planes The intersection of these two planes is shown by Line AB. A B Try to imagine how two planes could intersect at a point, or in any other way than a line. Various Planes Defined by 3 points Imagine or shade in Plane BAW in the below drawing. Slide 105 / 206
Various Planes Defined by 3 points Slide 106 / 206 Plane BAW What are the 3 other ways you can name this same plane? Various Planes Defined by 3 points Slide 107 / 206 Imagine or shade in Plane AZW in the below drawing. Various Planes Defined by 3 points Plane AZW What are the 3 other ways you can name this same plane? Slide 108 / 206
Various Planes Defined by 3 points Slide 109 / 206 Draw Plane UYA in the below drawing. Various Planes Defined by 3 points Slide 110 / 206 Plane UYA What are the 3 other ways you can name this same plane? Various Planes Defined by 3 points Imagine or draw Plane ABU in the below drawing. Slide 111 / 206
Slide 112 / 206 Various Planes Defined by 3 points Plane ABU What are the 3 other ways you can name this same plane? Slide 113 / 206 A D C B A D Answer 27 Name the point that is not in plane ABC. B C Slide 114 / 206 A D C B A B C D Answer 28 Name the point that is not in plane DBC.
Slide 115 / 206 29 Name two points that are in both indicated planes. A D B Answer AD B C C Slide 116 / 206 30 Name two points that are not on Line BC. A D B Answer AD B C C Slide 117 / 206 Yes No Answer 31 Line BC does not contain point R. Are points R, B, and C collinear? Draw the situation if it helps.
Slide 118 / 206 32 Plane LMN does not contain point P. Are points P, M, and N coplanar? No Answer Yes Slide 119 / 206 33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture) No Answer Yes Slide 120 / 206 34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar? No Answer Yes
Slide 121 / 206 35 Line BA and line DB intersect at Point . Answer A DH E G B F C Slide 122 / 206 A E, F, B, A B A, C, G, E C D, H, G, C D F, E, G, H Answer 36 Which group of points are noncoplanar with points A, B, and F on the cube below. Slide 123 / 206 37 Are lines EF and CD coplanar on the cube below? Yes Answer No
Slide 124 / 206 A C B line DC C Line CG D they don't intersect Answer 38 Plane ABC and plane DCG intersect at ? Slide 125 / 206 39 Planes ABC, GCD, and EGC intersect at ? A line GC C point A D line AC Answer B point C Slide 126 / 206 40 Name another point that is in the same plane as AF D BG E CH Answer points E, G, and H.
41 Name a point that is coplanar with points E, F, and C. Answer AF D BG E CH Slide 127 / 206 Slide 128 / 206 A Always B Sometimes C Never Answer 42 Intersecting lines are coplanar. Slide 129 / 206 A Always B Sometimes C Never Answer 43 Two planes intersect atexactly one point.
Slide 130 / 206 A Always B Sometimes C Never Answer 44 A plane can be drawn so that any three points are coplaner Slide 131 / 206 A Always B Sometimes C Never Answer 45 A plane containing two points of a line contains the entire line. Slide 132 / 206 A Always B Sometimes C Never Answer 46 Four points are noncoplanar.
Slide 133 / 206 A Always B Sometimes C Never Answer 47 Two lines meet at more than one point. Slide 134 / 206 Congruence, Distance and Length Return to Table of Contents Congruence 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: a b which is read as "a is congruent to b." Slide 135 / 206
Slide 136 / 206 Congruence 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. Slide 137 / 206 Congruence 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. b a Congruence 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. a b Slide 138 / 206
Slide 139 / 206 Congruence Would the same be true for any two rays? a b Slide 140 / 206 Congruence Again, all rays are infinitely long, so they have the same length. And once their vertices and any other point on both rays overlap, all of their points will overlap. All rays are congruent. b a Slide 141 / 206 Congruence Would the same be true of all line segments? a b
Slide 142 / 206 Congruence 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. b a Distance and Length Slide 143 / 206 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: Length: Distance and Length 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. Slide 144 / 206
Slide 145 / 206 Distance and Length 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. A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 146 / 206 Distance and Length 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. A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 147 / 206 Distance and Length 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. A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Slide 148 / 206 Distance and Length Any line segment which has a length of 7 will be congruent with CE, even if it needs to be rotated or moved to overlap it. All such segments have the same length regardless of orientation. So, segment CE and EC are congruent and have length 7. A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 149 / 206 Distance and Length What is the distance of the line below? Is that answer positive or negative? A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 150 / 206 Distance and Length All measures of distance
Points, Lines, Planes & Angles Part 1 www.njctl.org 2014-09-05 Slide 2 / 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 .
2b 2 points 3 2 points 4 6 points 5 4 points 6 2 points 7 4 points 8 3 points subtotal / 26 Large Numbers and Place Value 9 3 points 10 2 points 11 3 points 12 3 points 13 2 points 14 3 points 15 3 points 16 4 points subtotal / 23 Multi-Digit Multiplication 17 6 points 18 3 points 19 8 points 20 3 points 21a 3 points 21b 2 points
Points, Lines, Planes, & Angles Points & Lines: CLASSWORK 1. Name three collinear points on line q and on line s 2. Name 4 sets of non-collinear points 3. Name the opposite rays on line q and on line s 4. How many points are marked on line q? 5. How many points are there on line q? Points & Lines: HOMEWORK 6.
Chapter 1: Tools of Geometry Chapter 1 – Day 1 – Points, Lines and Planes Objectives: SWBAT identify Points, Lines, Rays, and Planes. SWBAT identify Coplanar and Non-Coplanar Points. Point The basic unit of Geometry. It is a location Line An infinite series of points extending in two opposite directions. Any two points make a line Plane
NAME 3 Spectrum Geometry Lesson 1.1 Grades 6-8 Points, Lines, Rays, and Angles Lesson 1.1 Points and Lines Collinear points are three or more points on the same straight line. Points that do not appear on the same straight line are noncollinear. A midpoint is the point halfway between the end points on a line segment. On the line WY at right, the midpoint is X.
www.ck12.orgChapter 1. Basics of Geometry, Answer Key CHAPTER 1 Basics of Geometry, Answer Key Chapter Outline 1.1 GEOMETRY - SECOND EDITION, POINTS, LINES, AND PLANES, REVIEW AN- SWERS 1.2 GEOMETRY - SECOND EDITION, SEGMENTS AND DISTANCE, REVIEW ANSWERS 1.3 GEOMETRY - SECOND EDITION, ANGLES AND MEASUREMENT, REVIEW AN- SWERS 1.4 GEOMETRY - SECOND EDITION, MIDPOINTS AND BISECTORS, REVIEW AN-
Computer Vision 2. The Projective Plane Motivation In Euclidean (planar) geometry, there are many exceptions, e.g., Imost two lines intersect in exactly one point. Ibut some two lines do not intersect. I parallel lines Idea: Iadd ideal points, one for each set of parallel lines / direction Ide ne these points as intersection of any two parallel lines Inow any two lines intersect in exactly one .
coplanar points/ lines These are po ints/ lines on the same plane. interesting lines Two or more lines are intersecting if they have a common point. parallel lines These are coplanar lines that do not meet. concurrent lines Three or more lines are concurrent if th ey all intersec
o Academic Writing , Stephen Bailey (Routledge, 2006) o 50 Steps to Improving your Academic Writing , Christ Sowton (Garnet, 2012) Complete introduction to organising and writing different types of essays, plus detailed explanations and exercises on sentence structure and linking: Writing Academic English , Alice
