Slide 28 / 209 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 29 / 209 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 30 / 209 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 31 / 209 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 32 / 209 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 33 / 209 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 34 / 209 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 35 / 209 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 36 / 209 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 37 / 209 Points and Lines Return to Table of Contents Slide 38 / 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 39 / 209 Points 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.
Slide 40 / 209 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 40 (Answer) / 209 Points Definition 1: A point is that which has no part. Math Practice Question on this slide addresses MP3 Look at this dot. Why can it not be considered a point? Discuss your answer with a partner. [This object is a pull tab] Slide 41 / 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). B A C
Slide 42 / 209 Lines 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. Slide 43 / 209 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 43 (Answer) / 209 Lines Definition 3: The ends of a line are points. Math Practice A line consists of an infinite number of points laid side by side, so Question on this slide addresses MP6 at either end of a line are points. & MP7 These are called endpoints. [This object is a pull tab] Even though this is how we correctly depict a line with endpoints, why is is not accurate?
Slide 44 / 209 Lines Definition 4. A straight line is a line which lies evenly with the points on itself. In a straight line the points lie next to one another without bending or turning in any direction. While a line can follow any path, in this course we will use the term "line" to mean a straight line, unless otherwise indicated. Slide 45 / 209 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 46 / 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.
Slide 47 / 209 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 48 / 209 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. Slide 48 (Answer) / 209 Naming Line Segments AB or BA A B endpoint endpoint MP6 Math Practice Remind students throughout this lesson about the proper notation and letter order A line segment is named by its two endpoints. (if required) for naming segments, rays & lines. The order of the endpoints doesn't matter. [This object is a pull tab] For instance, AB and BA are different names for the same segment.
Slide 49 / 209 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 50 / 209 Lines In this example, Line Segment AB is extended in both directions to create Line AB. A B endpoint endpoint B A Slide 51 / 209 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, F their order doesn't matter since the line Here are 7 valid goes in both E names for this line. directions. D DF EF DE FD FE a ED a
Slide 51 (Answer) / 209 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. Math Practice MP6 Remind students throughout this lesson When using two about the proper notation and letter order points to name a line, F their order doesn't (if required) for naming segments, rays & matter since the line lines. Here are 7 valid goes in both E names for this line. directions. D DF EF DE [This object is a pull tab] a ED FD FE a Slide 52 / 209 Example Give 7 different names for this line. b U V W Slide 52 (Answer) / 209 Example Give 7 different names for this line. b Answer U b V [This object is a pull tab] W
Slide 53 / 209 Collinear Points Collinear points are points which fall on the same line. Which of these points are collinear with the drawn line? B F C A E D a Slide 53 (Answer) / 209 Collinear Points Collinear points are points which fall on the same line. Which of these points are collinear with the drawn line? Points D, E, and F are Answer B collinear. C F A Points A, B, and C are not. E D [This object is a pull tab] a Slide 54 / 209 Collinear Points 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 (Answer) / 209 Collinear Points No, because a line can always be drawn between any two points. Is it possible for any two points to not be collinear on at least Answer one line? It's only when there are three or more points that they may not be collinear. Come up with an answer at your table. Remember, only use facts to make your argument! Questions/comments on this slide address MP3 & MP7 [This object is a pull tab] Slide 55 / 209 1 How many points are needed to define a line? Slide 55 (Answer) / 209 1 How many points are needed to define a line? Answer 2 points [This object is a pull tab]
Slide 56 / 209 2 Can there be two points which are not collinear on some line? Yes No Slide 56 (Answer) / 209 2 Can there be two points which are not collinear on some line? Yes Answer No No [This object is a pull tab] Slide 57 / 209 3 Can there be three points which are not collinear on some line? Yes No
Slide 57 (Answer) / 209 3 Can there be three points which are not collinear on some line? Yes No, each pair of points No can make up one line, Answer but if 3 or more points are not collinear, then they cannot be one the same line. [This object is a pull tab] Slide 58 / 209 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 58 (Answer) / 209 Intersecting Lines Is it possible for two different lines to intersect at more than one point? Math Practice These next 5 slides, including the current A good technique to prove whether this is possible is called either one address MP1, MP2 & MP3. Argumentum ad absurdum or Reductio ad absurdum [This object is a pull tab]
Slide 59 / 209 Intersecting Lines Argumentum ad absurdum or Reductio ad absurdum These are two Latin terms which refer to the same powerful approach, an indirect proof. First, you assume something is true. Then you see what logically follows from that assumption. If the conclusion is absurd, the assumption was false, and disproven. Slide 60 / 209 Intersecting Lines 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. Slide 61 / 209 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 62 / 209 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 63 / 209 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 T S Slide 64 / 209 4 What is the maximum number of points at which two distinct lines can intersect?
Slide 64 (Answer) / 209 4 What is the maximum number of points at which two distinct lines can intersect? Answer One intersection point [This object is a pull tab] Slide 65 / 209 5 Which sets of points are collinear on the lines drawn in this diagram? A A, D, B B C, D, B C A, D, C D none A D C B Slide 65 (Answer) / 209 5 Which sets of points are collinear on the lines drawn in this diagram? A A, D, B B C, D, B C A, D, C D none Answer C A D C [This object is a pull tab] B
Slide 66 / 209 6 At which point, or points, do the drawn lines intersect? A A and D B A and C C none D D A D C B Slide 66 (Answer) / 209 6 At which point, or points, do the drawn lines intersect? A A and D B A and C C none D D Answer D A D C [This object is a pull tab] B Slide 67 / 209 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 B endpoint endpoint A B
Slide 67 (Answer) / 209 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 On the next 4 slides, this one included: Math Practice create Ray AB. MP6 is addressed Remind students throughout this lesson about the proper notation and letter order A B (if required) for naming segments, rays & endpoint endpoint lines [This object is a pull tab] A B Slide 68 / 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? Line AB or Line BA A B Ray AB A B Slide 69 / 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. AB or BA B A AB B A
Slide 70 / 209 Naming Rays A B Segment AB can be extended in either in either direction. We can extend it at B to get ray AB. B A AB Or, we can extend it at A to get Ray BA. B A BA Slide 71 / 209 Naming Rays B A AB A B BA Rays AB and BA are NOT the same. What is the difference between them? Slide 71 (Answer) / 209 Naming Rays B A AB Math Practice Question on this slide addresses MP3 B A BA Rays AB and BA are NOT the same. What is the difference between them? [This object is a pull tab]
Slide 72 / 209 Opposite Rays 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. C B A Rays CA and CB are opposite rays . Slide 73 / 209 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 74 / 209 7 Name a point which is collinear with points G & H. A G E H D F B C C A B G D H E F
Slide 74 (Answer) / 209 7 Name a point which is collinear with points G & H. A G E H D F B C C Answer A A B G D H E F [This object is a pull tab] Slide 75 / 209 8 Name a point which is collinear with points D & A. A G E H D F B C C A B G D H E F Slide 75 (Answer) / 209 8 Name a point which is collinear with points D & A. A G E H D F B C C Answer A F B G D H E F [This object is a pull tab]
Slide 76 / 209 9 Name a point which is collinear with points D & E. A G E H D F B C C A B G D H E F Slide 76 (Answer) / 209 9 Name a point which is collinear with points D & E. A G E H D F B C C Answer A B B G D H E F [This object is a pull tab] Slide 77 / 209 10 Name a point which is collinear with points C & G. A G E H D F B C C A B G D H E F
Slide 77 (Answer) / 209 10 Name a point which is collinear with points C & G. A G E H D F B C C A Answer B E G D H E F [This object is a pull tab] Slide 78 / 209 11 Name an opposite ray to Ray MN. A Ray MQ B Ray MO C Ray RO D Ray PR O M N P Q R S T Slide 78 (Answer) / 209 11 Name an opposite ray to Ray MN. A Ray MQ B Ray MO Answer B C Ray RO D Ray PR O M N [This object is a pull P Q tab] R S T
Slide 79 / 209 12 Name an opposite ray to Ray PS. A Ray MQ B Ray MO C Ray PO D Ray PR O M N P Q R S T Slide 79 (Answer) / 209 12 Name an opposite ray to Ray PS. A Ray MQ B Ray MO C Ray PO D Ray PR Answer C O M N P Q [This object is a pull tab] R S T Slide 80 / 209 13 Name an opposite ray to Ray PM. A Ray MQ B Ray MO C Ray PO D Ray PR O M N P Q R S T
Slide 80 (Answer) / 209 13 Name an opposite ray to Ray PM. A Ray MQ B Ray MO C Ray PO D Ray PR Answer D O M N [This object is a pull P Q tab] R S T Slide 81 / 209 14 Rays HE and HF are the same. True False p P D E H F G g Slide 81 (Answer) / 209 14 Rays HE and HF are the same. True False p Answer P False D E H [This object is a pull tab] F G g
Slide 82 / 209 15 Rays HE and HP are the same. True False p P D E H F G g Slide 82 (Answer) / 209 15 Rays HE and HP are the same. True False p Answer P True D E H [This object is a pull F G tab] g Slide 83 / 209 16 Lines EH and EF are the same. True False p P D E H F G g
Slide 83 (Answer) / 209 16 Lines EH and EF are the same. True False p Answer P True D E H [This object is a pull F G tab] g Slide 84 / 209 17 Line p contains just three points. True False p P D E H F G g Slide 84 (Answer) / 209 17 Line p contains just three points. True False p Answer P False D E H [This object is a pull F tab] G g
Slide 85 / 209 18 Points D, H, and E are collinear. True False p P D E H F G g Slide 85 (Answer) / 209 18 Points D, H, and E are collinear. True False p Answer P False D E H [This object is a pull F tab] G g Slide 86 / 209 19 Points G, D, and H are collinear. True False p P D E H F G g
Slide 86 (Answer) / 209 19 Points G, D, and H are collinear. True False p Answer P True D E H [This object is a pull tab] F G g Slide 87 / 209 20 Are ray LJ and ray JL opposite rays? Yes No J K L Slide 87 (Answer) / 209 20 Are ray LJ and ray JL opposite rays? Yes No No, opposite rays have Answer same endpoint but point in opposite directions. J K [This object is a pull tab] L
Slide 88 / 209 21 Which of the following are opposite rays? A ray JK & ray LK C ray KJ & ray KL B ray JK & ray LK D ray JL & ray KL J K L Slide 88 (Answer) / 209 21 Which of the following are opposite rays? A ray JK & ray LK C ray KJ & ray KL B ray JK & ray LK D ray JL & ray KL Answer C J K [This object is a pull tab] L Slide 89 / 209 22 Name the initial point of ray AC. A A B C B C
Slide 89 (Answer) / 209 22 Name the initial point of ray AC. A A B Answer A C B C [This object is a pull tab] Slide 90 / 209 23 Name the initial point of ray BC. A A B C B C Slide 90 (Answer) / 209 23 Name the initial point of ray BC. A A B Answer B C B [This object is a pull C tab]
Slide 91 / 209 Planes Return to Table of Contents Slide 92 / 209 Planes 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. Slide 93 / 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 (Answer) / 209 Planes Question on this slide addresses MP7. Math Practice Recall that points which fall on the same line are called collinear points. Additional questioning, if needed: What does collinear mean? (MP6) With that in mind, what do you think points on the same How is collinear related to coplanar? plane are called? (MP7) [This object is a pull tab] Slide 94 / 209 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 95 / 209 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?
Slide 95 (Answer) / 209 Planes Definition 7: A plane surface is a surface which lies evenly with the straight lines on itself. Question on this slide addresses MP7. Math Practice Additional questioning, if needed: This indicates that the surface of the plane is flat so that What does point mean? (MP6) lines on the plane will lie flat on it. What does line mean? (MP6) How are points and lines related to Thinking about the definitions of points and lines, exactly planes? (MP7) how flat do you think a plane is? [This object is a pull tab] Slide 96 / 209 Coplanar Points and Lines 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 Slide 97 / 209 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 98 / 209 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 99 / 209 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 (Answer) / 209 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 Math Practice which they are coplanar? Questions on this slide address MP2. Can you draw it on the image? What could be a name for that plane? [This object is a pull tab]
Slide 100 / 209 Coplanar Points 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. Slide 100 (Answer) / 209 Coplanar Points No, because a plane can Is it possible for any always be drawn which three points to not be contains any three points. coplanar with one Answer another? It's only when there are four or more points that they may Try and find 3 points not be coplanar. on this diagram Questions on this slide address which are not coplanar. MP2 & MP3. [This object is a pull tab] Slide 101 / 209 24 How many points are needed to define a plane?
Slide 101 (Answer) / 209 24 How many points are needed to define a plane? Answer 3 points [This object is a pull tab] Slide 102 / 209 25 Can there be three points which are not coplanar on any plane? Yes No Slide 102 (Answer) / 209 25 Can there be three points which are not coplanar on any plane? Yes No Answer No [This object is a pull tab]
Slide 103 / 209 26 Can there be four points which are not coplaner on any plane? Yes No Slide 103 (Answer) / 209 26 Can there be four points which are not coplaner on any plane? Yes No, if the 4 points are No Answer not coplanar, then one of the points is not in the same plane as the other 3. [This object is a pull tab] Slide 104 / 209 Intersecting Planes What would the intersection of two planes look like? Hint: the walls and ceiling of this room could represent planes.
Slide 104 (Answer) / 209 Intersecting Planes The intersection of any two planes is a line. What would the intersection of two planes look like? Questions on this slide address MP2 Answer Hint: the walls and ceiling of this room could represent planes. & 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) [This object is a pull tab] Slide 105 / 209 Intersecting Planes The intersection of these two planes is shown by Line AB. B A Try to imagine how two planes could intersect at a point, or in any other way than a line. Slide 106 / 209 Various Planes Defined by 3 points Imagine or shade in Plane BAW in the below drawing.
Slide 107 / 209 Various Planes Defined by 3 points Plane BAW What are the 3 other ways you can name this same plane? Slide 107 (Answer) / 209 Various Planes Defined by 3 points Plane BAW Plane XBA, XWA, BXW, or the reordering of any of Answer these letter combinations What are the 3 other ways you can name this same plane? Question on this slide addresses MP7 [This object is a pull tab] Slide 108 / 209 Various Planes Defined by 3 points Imagine or shade in Plane AZW in the below drawing.
Slide 109 / 209 Various Planes Defined by 3 points Plane AZW What are the 3 other ways you can name this same plane? Slide 109 (Answer) / 209 Various Planes Defined by 3 points Plane WVZ, WAZ, AWV, or Plane AZW the reordering of any of Answer these letter combinations Question on this slide addresses MP7 What are the 3 other ways you can name this same plane? [This object is a pull tab] Slide 110 / 209 Various Planes Defined by 3 points Draw Plane UYA in the below drawing.
Slide 111 / 209 Various Planes Defined by 3 points Plane UYA What are the 3 other ways you can name this same plane? Slide 111 (Answer) / 209 Various Planes Defined by 3 points Plane UYA Plane UWA, WAY, WUY, or the reordering of any of Answer these letter combinations What are the 3 other ways you can name this same plane? Question on this slide addresses MP7 [This object is a pull tab] Slide 112 / 209 Various Planes Defined by 3 points Imagine or draw Plane ABU in the below drawing.
Slide 113 / 209 Various Planes Defined by 3 points Plane ABU What are the 3 other ways you can name this same plane? Slide 113 (Answer) / 209 Various Planes Defined by 3 points Plane ABU Plane AVU, BUV, BAV, or the reordering of Answer any of these letter combinations What are the 3 other ways you can name this Question on this slide addresses same plane? MP7 [This object is a pull tab] Slide 114 / 209 27 Name the point that is not in plane ABC. A D C B A D B C
Slide 114 (Answer) / 209 27 Name the point that is not in plane ABC. A D C B A Answer D D B C [This object is a pull tab] Slide 115 / 209 28 Name the point that is not in plane DBC. A D C B A D B C Slide 115 (Answer) / 209 28 Name the point that is not in plane DBC. A D C B A Answer D A B C [This object is a pull tab]
Slide 116 / 209 29 Name two points that are in both indicated planes. A D B C A D B C Slide 116 (Answer) / 209 29 Name two points that are in both indicated planes. A D B C A Answer D B & C B C [This object is a pull tab] Slide 117 / 209 30 Name two points that are not on Line BC. A D B C A D B C
Slide 117 (Answer) / 209 30 Name two points that are not on Line BC. A D B C A Answer D A & D B C [This object is a pull tab] 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 Slide 118 (Answer) / 209 31 Line BC does not contain point R. Are points R, B, and C collinear? Draw the situation if it helps. Answer Yes No No [This object is a pull tab]
Slide 119 / 209 32 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No Slide 119 (Answer) / 209 32 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No Answer Yes on Plane MNP [This object is a pull tab] Slide 120 / 209 33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture) Yes No
Slide 120 (Answer) / 209 33 Plane QRS contains line QV. Are points Q, R, S, and V coplanar? (Draw a picture) Yes No Answer Yes [This object is a pull tab] Slide 121 / 209 34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar? Yes No Slide 121 (Answer) / 209 34 Plane JKL does not contain line JN. Are points J, K, L, and N coplanar? Yes No Answer No [This object is a pull tab]
Slide 122 / 209 35 Line BA and line DB intersect at Point ____. A H D E G B F C Slide 122 (Answer) / 209 35 Line BA and line DB intersect at Point ____. A H D E G B F C Answer B [This object is a pull tab] Slide 123 / 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 (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 Answer C D, H, G, C C D F, E, G, H [This object is a pull tab] Slide 124 / 209 37 Are lines EF and CD coplanar on the cube below? Yes No Slide 124 (Answer) / 209 37 Are lines EF and CD coplanar on the cube below? Yes No Yes [This object is a pull tab]
Slide 125 / 209 38 Plane ABC and plane DCG intersect at _____? A C B line DC C Line CG D they don't intersect Slide 125 (Answer) / 209 38 Plane ABC and plane DCG intersect at _____? A C B line DC C Line CG Answer B D they don't intersect [This object is a pull tab] Slide 126 / 209 39 Planes ABC, GCD, and EGC intersect at _____? A line GC B point A C point C D line AC
Slide 126 (Answer) / 209 39 Planes ABC, GCD, and EGC intersect at _____? A line GC B point A C point C D line AC Answer C [This object is a pull tab] Slide 127 / 209 40 Name another point that is in the same plane as points E, G, and H. F A D G B E H C Slide 127 (Answer) / 209 40 Name another point that is in the same plane as points E, G, and H. F F A D Answer B G E C H [This object is a pull tab]
Slide 128 / 209 41 Name a point that is coplanar with points E, F, and C. A F D B G E H C Slide 128 (Answer) / 209 41 Name a point that is coplanar with points E, F, and C. F A D Answer G B D E H C [This object is a pull tab] Slide 129 / 209 42 Intersecting lines are __________ coplanar. A Always B Sometimes C Never
Slide 129 (Answer) / 209 42 Intersecting lines are __________ coplanar. A Always B Sometimes C Never Answer A [This object is a pull tab] Slide 130 / 209 43 Two planes ____________ intersect at exactly one point. A Always B Sometimes C Never Slide 130 (Answer) / 209 43 Two planes ____________ intersect at exactly one point. A Always B Sometimes Answer C Never C [This object is a pull tab]
Slide 131 / 209 44 A plane can __________ be drawn so that any three points are coplaner. A Always B Sometimes C Never Slide 131 (Answer) / 209 44 A plane can __________ be drawn so that any three points are coplaner. A Always B Sometimes Answer C Never A [This object is a pull tab] Slide 132 / 209 45 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes C Never
Slide 132 (Answer) / 209 45 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes Answer A C Never [This object is a pull tab] Slide 133 / 209 46 Four points are ____________ noncoplanar. A Always B Sometimes C Never Slide 133 (Answer) / 209 46 Four points are ____________ noncoplanar. A Always B Sometimes C Never Answer B [This object is a pull tab]
Slide 134 / 209 47 Two lines ________________ meet at more than one point. A Always B Sometimes C Never Slide 134 (Answer) / 209 47 Two lines ________________ meet at more than one point. A Always B Sometimes Answer C Never B [This object is a pull tab] Slide 135 / 209 Congruence, Distance and Length Return to Table of Contents
Slide 136 / 209 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 137 / 209 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 138 / 209 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
Slide 139 / 209 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 140 / 209 Congruence Would the same be true for any two rays? a b Slide 140 (Answer) / 209 Congruence Would the same be true for any two rays? Math Practice Questions on this slide address MP3. a b [This object is a pull tab]
Slide 141 / 209 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 142 / 209 Congruence Would the same be true of all line segments? a b Slide 142 (Answer) / 209 Congruence Would the same be true of all line segments? Math Practice Questions on this slide address MP3. a b [This object is a pull tab]
Slide 143 / 209 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 Slide 144 / 209 Distance and Length 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: Slide 144 (Answer) / 209 Distance and Length 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. Math Practice Questions on this slide address MP7. Distance: Length: [This object is a pull tab]
Slide 145 / 209 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 146 / 209 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 -9 -8 -7 -6 -5 -4 -3 0 2 3 4 5 6 7 8 9 10 -10 -2 -1 1 Slide 147 / 209 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 148 / 209 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 0 2 3 4 5 6 7 8 9 10 -2 -1 1 Slide 149 / 209 Distance and Length 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. 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 / 209 Distance and Length What is the distance of the line below? Is that answer positive or negative? B D E A C F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Slide 151 / 209 Distance and Length 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. A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 0 2 3 4 5 6 7 8 9 10 -2 -1 1 Slide 152 / 209 Distance 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? 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 152 (Answer) / 209 Distance 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 displacement to take to get from one to the other. ( X ) Which direction you walk along the line doesn't change the distance. Answer 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? Question addresses MP7 [This object is a pull tab] 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 153 / 209 48 What is the location of point F? A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 0 2 3 4 5 6 7 8 9 10 -2 -1 1 Slide 153 (Answer) / 209 48 What is the location of point F? Answer +10 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 [This object is a pull tab] Slide 154 / 209 49 What is the location of point A? D A B C E F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Slide 154 (Answer) / 209 49 What is the location of point A? Answer -7 A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 0 2 3 4 5 6 7 8 9 10 -2 -1 1 [This object is a pull tab] Slide 155 / 209 50 What is the distance from A to C? A B C D E F -10 -9 -8 -7 -6 -5 -4 -3 0 2 3 4 5 6 7 8 9 10 -2 -1 1 Slide 155 (Answer) / 209 50 What is the distance from A to C? Answer 7 units A B C D E F [This object is a pull tab] -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Slide 156 / 209 51 What is the distance from B to E? 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 156 (Answer) / 209 51 What is the distance from B to E? Answer 11 units A B C D E F [This object is a pull tab] -10 -9 -8 -7 -6 -5 -4 -3 0 2 3 4 5 6 7 8 9 10 -2 -1 1 Slide 157 / 209 52 What is the distance from B to A? 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 157 (Answer) / 209 52 What is the distance from B to A? Answer 3 units A B C D E F [This object is a pull tab] -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 158 / 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 of which direction you drive...only how far you drive matters. Slide 159 / 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. 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 160 / 209 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. B D E A C F -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 161 / 209 53 What's the distance between A and F? A B C D E F -9 -8 -7 -6 -5 -4 -3 2 3 4 5 6 7 8 9 10 -10 -2 -1 0 1 Slide 161 (Answer) / 209 53 What's the distance between A and F? Answer 17 units [This object is a pull tab] 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 162 / 209 54 What's the distance between two points if one is located at +125 and the other is located at -350? Slide 162 (Answer) / 209 54 What's the distance between two points if one is located at +125 and the other is located at -350? Answer 475 units [This object is a pull tab] Slide 163 / 209 55 What's the distance between two points if one is located at -540 and the other is located at -180?
Slide 163 (Answer) / 209 55 What's the distance between two points if one is located at -540 and the other is located at -180? Answer 360 units [This object is a pull tab] Slide 164 / 209 Example Find the measure of each segment in centimeters. A B C E D cm F a. CE = 8 - 2 = 6 cm b. AB = 1.5 cm Note: When giving the measurement of segments using an equal sign, the segment bar is not used. Slide 165 / 209 56 Find a segment that is 4 cm long. A B C D A B C D E cm F
Slide 165 (Answer) / 209 56 Find a segment that is 4 cm long. A B C Answer D B A B C D E [This object is a pull tab] cm F Slide 166 / 209 57 Find a segment that is 6.5 cm long. A B C D A B C D E cm F Slide 166 (Answer) / 209 57 Find a segment that is 6.5 cm long. A B C Answer D B A B C D E [This object is a pull tab] cm F
Slide 167 / 209 58 Find a segment that is 3.5 cm long. A B C D A B C E D cm F Slide 167 (Answer) / 209 58 Find a segment that is 3.5 cm long. A B C D Answer D A B C D E [This object is a pull tab] cm F Slide 168 / 209 59 Find a segment that is 2 cm long. A B C D A B C D E cm F
Slide 168 (Answer) / 209 59 Find a segment that is 2 cm long. A B C D Answer B A B C D E [This object is a pull tab] cm F Slide 169 / 209 60 Find a segment that is 5.5 cm long. A B C D A B C D E cm F Slide 169 (Answer) / 209 60 Find a segment that is 5.5 cm long. A B C Answer D D A B C D E [This object is a pull tab] cm 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 A B C D E cm F Slide 170 (Answer) / 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 Answer C 3.5 cm D D 4.5 cm A B C D E [This object is a pull tab] cm F Slide 171 / 209 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. A B C AB BC AC
Slide 172 / 209 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. A B C AB BC AC Slide 173 / 209 Adding Line Segments This works for any number of segments on a line. A B C D E AB + BC + CD + DE = AE Slide 174 / 209 Example D E A B C Given: AE = 27 AB = CD DE = 5 BC= 6 Find: BE CD
Slide 174 (Answer) / 209 Example create a variable for the 2 unknown equivalent lengths, x D E A B C AB = CD = x AB + BC + CD + DE = AE x + 6 + x + 5 = 27 Answer Given: AE = 27 2x + 11 = 27 2x = 16 AB = CD x = 8 units = CD DE = 5 BC= 6 BE = 6 + 8 + 5 = 19 units This example addresses MP1 & MP2 Find: BE [This object is a pull tab] CD Slide 175 / 209 Example Label the line and find the value of x given that: P lies between K and M on a line. PM= 2x + 4 MK= 14x - 56 PK = x + 17 Slide 175 (Answer) / 209 Example Label the line and find the value of x given that: Solve for x P lies between K and M on a line. (x + 17) + (2x + 4) = 14x - 56 3x + 21 = 14x - 56 + 56 + 56 PM= 2x + 4 Answer MK= 14x - 56 PK = x + 17 3x + 77 = 14x -3x - 3x 77 = 11x 7 = x This example addresses MP 1 & MP2 [This object is a pull tab]
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 Slide 176 (Answer) / 209 Example P, B, L, and M are collinear and are in the following order: 8x + 40 = 3x + 140 a) P is between B and M 5x + 40 = 140 Answer b) L is between M and P 5x = 100 x = 20 Draw a diagram and solve for x, given: This example addresses MP1 & MP2 ML = 3x +16 [This object is a pull tab] PL = 2x +11 BM = 3x +140 PB = 3x + 13 Slide 177 / 209 62 What is the length of Segment AB? A B C D E Hint: always start these problems by placing the information you have into the diagram.
Slide 177 (Answer) / 209 62 What is the length of Segment AB? A B C D E Answer 3 Hint: always start these problems [This object is a pull tab] by placing the information you have into the diagram. Slide 178 / 209 63 What is the length of Segment DE? A B C D E Slide 178 (Answer) / 209 63 What is the length of Segment DE? D E A B C Answer 11 [This object is a pull tab]
Slide 179 / 209 64 What is the length of Segment CA? D E A B C Slide 179 (Answer) / 209 64 What is the length of Segment CA? E A B C D Answer 6 [This object is a pull tab] Slide 180 / 209 65 What is the length of Segment CE? D E A B C
Slide 180 (Answer) / 209 65 What is the length of Segment CE? D E A B C Answer 14 [This object is a pull tab] Slide 181 / 209 66 What is the length of Segment CE? A B C D E Slide 181 (Answer) / 209 66 What is the length of Segment CE? D E A B C Answer 14 [This object is a pull tab]
Slide 182 / 209 67 What is the length of Segment DA? E A B C D Slide 182 (Answer) / 209 67 What is the length of Segment DA? A B C D E Answer 9 [This object is a pull tab] Slide 183 / 209 68 What is the length of Segment BE? E A B C D
Slide 183 (Answer) / 209 68 What is the length of Segment BE? A B C D E Answer 17 [This object is a pull tab] 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 B X Slide 184 (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 x - 12 + 15 x -7 = 6 x + 151 XY = 15x - 7 16 x - 19 = 6 x + 151 10 x = 170 Answer BY = x - 12 x = 17 15 x - 7 x - 12 B Y X [This object is a pull 6 x + 151 tab] B X Y
Slide 185 / 209 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 R X Q Slide 185 (Answer) / 209 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 7 x + 1 + 15 x + 10 = 2 x + 131 22 x + 11 = 2 x + 131 RQ = 2x + 131 20 x = 120 Answer x = 6 XR = 7x +1 15 x + 10 7 x + 1 X R Q 2 x + [This object is a pull tab] 131 R Q X Slide 186 / 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 B K V
Slide 186 (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 4 x + 149 + 5 x = 15 x + 125 9 x + 149 = 15 x + 125 BV = 15x + 125 Answer 6 x = 24 KV = 4x +149 x = 4 5 x 4 x + 149 K V B 15 x + 125 [This object is a pull tab] B K V Slide 187 / 209 Constructions and Loci Return to Table of Contents Slide 187 (Answer) / 209 Constructions Math Practice This entire lesson w/ constructions addresses MP5 and Loci [This object is a pull tab] Return to Table of Contents
Slide 188 / 209 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 189 / 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 r learn a bit now so we can proceed with constructions. Slide 190 / 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 191 / 209 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 192 / 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 193 / 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 194 / 209 Introduction to Constructions 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. r center circle Slide 195 / 209 Try this! 1) Create a circle using the segment below. E M F Slide 195 (Answer) / 209 Try this! 1) Create a circle using the segment below. The file for the "Try This!" E problems is located on the NJCTL website: Teacher Notes https://njctl.org/courses/math/ geometry-2015-16/points-lines- and-planes/constructions- M worksheet-for-presentation/ Called "Constructions Worksheet" in the "Handouts" section. [This object is a pull F tab]
Slide 196 / 209 Try this! 2) Create a circle using the segment below. H M G Slide 197 / 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. A B Slide 198 / 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. A B C a
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