Geometry. Points, Lines, Planes & Angles. Part 1. Introduction to Geometry. Slide 1 / 206 Slide 2 / 206. Slide 3 / 206. Slide 4 / 206.

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1 Slide 1 / 206 Slide 2 / 206 Geometry Points, Lines, Planes & ngles Part Slide 3 / 206 Slide 4 / 206 Part 1 Introduction to Geometry Table of ontents Points and Lines Planes ongruence, istance and Length onstructions and Loci Part 2 ngles ongruent ngles ngles & ngle ddition Postulate Protractors Special ngle Pairs Proofs Special ngles ngle isectors Locus & ngle onstructions ngle isectors & onstructions click on the topic to go to that section Introduction to Geometry Return to Table of ontents Slide 5 / 206 Slide 6 / 206 The Origin of Geometry bout 10,000 years ago much of North frica was fertile farmland. The area around the Nile river was too marshy for agriculture, so it was sparsely populated. The Origin of Geometry ut over thousands of years the climate changed, and most of North frican became desert. The banks of the Nile became prime farmland.

2 Slide 7 / 206 The Origin of Geometry The land along the Nile became crowded with people. Slide 8 / 206 gyptian Geometry bout 4000 years ago an gyptian pharaoh, Sesostris, is said to have invented geometry in order to keep track of the land and tax it's owners. arming was done on the land near the river because it had: Water for irrigation ertile soil due to annual flooding, which deposited silt from upriver. ut, since the land flooded each year, how could they keep track of who owned which land? Reestablishing land ownership after each annual flood required a practical geometry. "Geo" means arth and "metria" means measure, so geometry meant to measure land. Slide 9 / 206 Land oundaries Lab Slide 10 / 206 Land oundaries Lab Pre- lood oundary Map You know more geometry than the gyptians knew 4000 years ago, so let's do a lab to see how you would solve this problem. efore the annual flood of the Nile three plots of land might be as shown. Plot 1 You'll work in groups and each group will solve this problem before we move on to how the Greek's built on the gyptian solution. 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. Plot 2 Plot 3 Slide 11 / 206 Land oundaries Lab Slide 12 / 206 Land oundaries Lab efore flooding, three plots of land might be look like these. Pre- lood oundary Map fterwards, only the stakes above the flood level remained, and the river had moved in its course. Post-lood Map of River and Markers The pharaoh had to: Reestablish new boundaries so farmers knew which land to farm. djust the taxes to match the new amount of land owned. Plot 1 Plot 2 Plot 3 The gyptians only had stakes and rope, you only have tape and string.

3 Slide 13 / 206 Land oundaries Lab Slide 14 / 206 gyptian Geometry fter the flood, the pharaoh would send out geometers with ropes that had been used to measure each plot of land in prior years. How did they do it? (You can't use the edges of the paper or rulers because these were open fields of great size.) Post-lood Map of River and Markers gyptian mathematics was very practical. What practical applications do you think the gyptians used mathematics for? They did not develop abstract mathematics. That was left to the Greeks, who built upon what they had learned from the gyptians, abylonians and others. Slide 15 / 206 Slide 16 / 206 Greek Geometry uclidean 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. uclid's book, The lements, summarized the results of Greek geometry: uclidean Geometry. uclidean geometry is the basis of much of western mathematics, philosophy and science. It also represents a great place to learn that type of thinking. Slide 17 / 206 uclidean Geometry uclidean Geometry dates prior to 400. 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 uclid's ideas? This statement was posted above Plato's cademy, in ancient thens, about 2500 years ago. This renaissance painting by Raphael depicts that academy. Slide 18 / 206 uclidean Geometry "Let none who are ignorant of geometry enter here."

4 Slide 19 / 206 uclidean Geometry Slide 20 / 206 uclidean 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 lements by uclid. uclidean Geometry was lost to urope for a 1000 years. ut, it continued to be used and developed in the Islamic world. In the 1400's, these ideas were reintroduced to urope. These, and other rediscovered works, led to the uropean Renaissance, which lasted several centuries, beginning in the 1400's. These works were not purposely destroyed, but deteriorated with age as there was no central government to maintain them. Much of the thinking of modern science and mathematics developed from the rediscovery of uclid's lements. Slide 21 / 206 uclidean Geometry Slide 22 / 206 uclidean Geometry bout 100 years ago, harles odgson, the Oxford geometer who wrote lice in Wonderland, under the name Lewis arroll, argued uclid was still the best way to understand mathematical thinking. The thinking that underlies uclidean Geometry has held up very well. Many still believe it is the best introduction to analytical thinking. Slide 23 / 206 uclidean Geometry Slide 24 / 206 uclidean 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 (STM). This course will use the basic thinking developed by uclid. 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. lways question every idea that's presented, that's what uclid and those who invented geometry would have wanted.

5 Slide 25 / 206 uclidean Geometry Slide 26 / 206 uclidean Geometry This also represents a path to logical thinking, which ritish philosopher ertrand Russell showed is identical to mathematical thinking. lick on the image to watch a short video of ertrand Russell's message to the future which was filmed in uclid'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 orollaries. id you hear anything that sounded familiar? What was it? Slide 27 / 206 uclidean 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. lso, additional modern terms which you will need to know will be introduced as needed. Slide 28 / 206 uclid's xioms (ommon Understandings) uclid called his axioms "ommon 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 Slide 30 / 206 uclid's xioms (ommon 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. or example: uclid's irst xiom Things which are equal to the same thing are also equal to one another. if I know that Tom and ob are the same height, and I know that ob and Sarah are the same height...what other conclusion can I come to? Without recognizing the assumptions you are making, you're not able to question them...and, sometimes, not able to move beyond them. Tom ob Sarah

6 Slide 31 / 206 uclid's Second xiom If equals are added to equals, the whole are equal. or 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 uclid's Third xiom If equals be subtracted from equals, the remainders are equal. This is just like the second axiom. ome up with an example on your own. Look back at the second axiom if you need a hint. Slide 33 / 206 uclid's ourth xiom Things which coincide with one another are equal to one another. Slide 34 / 206 uclid's ifth xiom The whole is greater than the part. or 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. or 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 Slide 36 / 206 uclid's xioms (ommon Understandings) irst xiom: Things which are equal to the same thing are also equal to one another. Second xiom: If equals are added to equals, the whole are equal. Points and Lines Third xiom: If equals be subtracted from equals, the remainders are equal. ourth xiom: Things which coincide with one another are equal to one another. ifth xiom: The whole is greater than the part. Return to Table of ontents

7 Slide 37 / 206 efinitions Slide 38 / 206 Points efinition 1: point is that which has no part. efinitions 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. 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 39 / 206 Points efinition 1: point is that which has no part. point is represented by a dot. Slide 40 / 206 Points The dot drawn on a page has dimensions, but the point it represents does not. point can be imagined, but not drawn. Look at this dot. Why can it not be considered a point? iscuss your answer with a partner. Only the position of the point is shown by the dot. Points are usually labeled with a capital letter (e.g.,, ). Slide 41 / 206 Lines efinition 2: line is breadthless length. Slide 42 / 206 Lines efinition 3: The ends of a line are points. 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. 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. ven though this is how we correctly depict a line with endpoints, why is is not accurate?

8 Slide 43 / 206 Lines efinition 4. straight line is a line which lies evenly with the points on itself. Slide 44 / 206 Lines irst Postulate: To draw a line from any point to any point. 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. This postulate indicates that given any two points, it is possible to draw a line between them. side 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. 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. line drawn in this way is called a line segment. It has finite length, a beginning and an end. t each end of the segment there is an endpoint, as shown below straight edge is a ruler without markings. Note: ny object with a straight edge can be used. endpoint endpoint Slide 47 / 206 Naming Line Segments Slide 48 / 206 Lines endpoint or endpoint line segment is named by its two endpoints. The order of the endpoints doesn't matter. 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. or instance, and are different names for the same segment.

9 Slide 49 / 206 Lines In this example, line Segment is extended in both directions to create Line. Slide 50 / 206 Naming Lines line is named by using any two points on it OR by using a single lower-case letter. rrowheads in the symbol above the points in the name of the line show that the line continues without end in opposite directions. endpoint endpoint When using two points to name a line, their order doesn't matter since the line goes in both directions. Here are 7 valid names for this line. a a b Slide 51 / 206 xample Give 7 different names for this line. Slide 52 / 206 ollinear Points ollinear points are points which fall on the same line. Which of these points are collinear with the drawn line? U V W a Slide 53 / 206 ollinear Points Slide 54 / How many points are needed to define a line? Is it possible for any two points to not be collinear on at least one line? ome up with an answer at your table. Remember, only use facts to make your argument!

10 Slide 55 / an there be two points which are not collinear on some line? Slide 56 / an there be three points which are not collinear on some line? Yes No Yes No Slide 57 / 206 Intersecting Lines Slide 58 / 206 Intersecting Lines Is it possible for two different lines to intersect at more than one point? rgumentum ad absurdum or good technique to prove whether this is possible is called either rgumentum ad absurdum or Reductio ad absurdum Reductio ad absurdum These are two Latin terms which refer to the same powerful approach, an indirect proof. irst, you assume something is true. Then you see what logically follows from that assumption. If the conclusion is absurd, the assumption was false, and disproven. Slide 59 / 206 Intersecting Lines Is it possible for two different lines to intersect at more than one point? Slide 60 / 206 Intersecting Lines Let's assume that two different lines can share more than one point and see where that leads us. Let's name the two points which are shared and. We could connect and with a line segment, since we can draw a line segment between any two points. That segment would overlap both our original lines between and, since they are all straight lines and all include and. We could then extend our Segment infinitely in both directions and our new Line 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. ut we assumed that the two original lines were different lines sharing two points.

11 Slide 61 / 206 Intersecting Lines Is it possible for two different lines to intersect at more than one point? Slide 62 / 206 Intersecting Lines Is it possible for two different lines to intersect at more than one point? ut we have concluded that they are the same line, not different lines. So, two different lines either: Intersect at no points It is impossible for them to be both different lines and the same lines. Intersect at one point. So, our assumption is proven false and the opposite assumption must be true. Two different lines cannot share two points. Q K R S T Slide 63 / What is the maximum number of points at which two distinct lines can intersect? Slide 64 / Which sets of points are collinear on the lines drawn in this diagram?,,,,,, none Slide 65 / t which point, or points, do the drawn lines intersect? and and none Slide 66 / 206 Rays 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. elow, the segment is extended to infinity, beyond Point, to create Ray. endpoint endpoint

12 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. Slide 68 / 206 Naming Rays lso, 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. Why do you think the order of the letters matter for rays? Line or Line or Ray Slide 69 / 206 Slide 70 / 206 Naming Rays Naming Rays Segment can be extended in either in either direction. We can extend it at to get ray. Or, we can extend it at to get Ray. Rays and are NOT the same. What is the difference between them? Slide 71 / 206 Opposite Rays Slide 72 / 206 ollinear Rays Opposite rays are defined as being two rays with a common endpoint that point in opposite directions and form a straight line. elow, suppose point is between points and. Recall: Since,, and all lie on the same line, we know they are collinear points. Rays and are opposite rays. Similarly, rays are also called collinear if they lie on the same line.

13 Slide 73 / Name a point which is collinear with points G & H. Slide 74 / Name a point which is collinear with points &. H G G H G G H H Slide 75 / 206 Slide 76 / Name a point which is collinear with points &. 10 Name a point which is collinear with points & G. H G G H G G H H Slide 77 / 206 Slide 78 / Name an opposite ray to Ray MN. Ray MQ Ray MO 12 Name an opposite ray to Ray PS. Ray MQ Ray MO Ray PO Ray PR Ray RO Ray PR O P M Q N O P M Q N R S T R S T

14 Slide 79 / Name an opposite ray to Ray PM. Ray MQ Ray MO Ray PO Ray PR Slide 80 / Rays H and H are the same. True alse p R O P S M Q T N H G P g Slide 81 / Rays H and HP are the same. True alse Slide 82 / Lines H and are the same. True alse p p P P H H G G g g Slide 83 / 206 Slide 84 / Line p contains just three points. 18 Points, H, and are collinear. True True alse p alse p P P H H G G g g

15 Slide 85 / Points G,, and H are collinear. True alse p Slide 86 / re ray LJ and ray JL opposite rays? Yes No P J H K G g L Slide 87 / Which of the following are opposite rays? Slide 88 / Name the initial point of ray. JK & LK JK & LK KJ & KL JL & KL J K L Slide 89 / 206 Slide 90 / Name the initial point of ray. Planes Return to Table of ontents

16 Slide 91 / 206 Planes Slide 92 / 206 Planes efinition 5: surface is that which has length and breadth only. 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. ut it has no height at all. Recall that points which fall on the same line are called collinear points. With that in mind, what do you think points on the same plane are called? Slide 93 / 206 Planes efinition 6: The edges of a surface are lines. Slide 94 / 206 Planes efinition 7: plane surface is a surface which lies evenly with the straight lines on itself. Just as the ends of lines are points, the edges of planes are lines. This indicates that the surface of the plane is flat so that lines on the plane will lie flat on it. Thinking about the definitions of points and lines, exactly how flat do you think a plane is? Slide 95 / 206 oplanar Points and Lines s you figured out earlier, coplanar points are points which fall on the same plane. Slide 96 / 206 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." a ll of the lines and points shown here are coplanar. lso, it can be named by the single letter, "Plane R."

17 Slide 97 / 206 oplanar Points Slide 98 / 206 oplanar Points oplanar points lie on the same plane. In this case, Points K, M, and L are coplanar and lie on the indicated plane. While points O, K, and L do not lie on the indicated plane, they are coplanar with one another. an you imagine a plane in which they are coplanar? an you draw it on the image? What could be a name for that plane? Slide 99 / 206 oplanar Points Slide 100 / How many points are needed to define a plane? 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 101 / an there be three points which are not coplanar on any plane? Slide 102 / an there be four points which are not coplaner on any plane? Yes No Yes No

18 Slide 103 / 206 Intersecting Planes Slide 104 / 206 Intersecting Planes What would the intersection of two planes look like? Hint: the walls and ceiling of this room could represent planes. The intersection of these two planes is shown by Line. Try to imagine how two planes could intersect at a point, or in any other way than a line. Slide 105 / 206 Various Planes efined by 3 points Imagine or shade in Plane W in the below drawing. Slide 106 / 206 Various Planes efined by 3 points Plane W What are the 3 other ways you can name this same plane? Slide 107 / 206 Various Planes efined by 3 points Imagine or shade in Plane ZW in the below drawing. Slide 108 / 206 Various Planes efined by 3 points Plane ZW What are the 3 other ways you can name this same plane?

19 Slide 109 / 206 Various Planes efined by 3 points raw Plane UY in the below drawing. Slide 110 / 206 Various Planes efined by 3 points Plane UY What are the 3 other ways you can name this same plane? Slide 111 / 206 Various Planes efined by 3 points Imagine or draw Plane U in the below drawing. Slide 112 / 206 Various Planes efined by 3 points Plane U What are the 3 other ways you can name this same plane? Slide 113 / Name the point that is not in plane. Slide 114 / Name the point that is not in plane.

20 Slide 115 / 206 Slide 116 / Name two points that are in both indicated planes. 30 Name two points that are not on Line. Slide 117 / 206 Slide 118 / Line does not contain point R. re points R,, and collinear? raw the situation if it helps. 32 Plane LMN does not contain point P. re points P, M, and N coplanar? Yes Yes No No Slide 119 / Plane QRS contains line QV. re points Q, R, S, and V coplanar? (raw a picture) Yes No Slide 120 / Plane JKL does not contain line JN. re points J, K, L, and N coplanar? Yes No

21 Slide 121 / Line and line intersect at Point. H G Slide 122 / Which group of points are noncoplanar with points,, and on the cube below.,,,,, G,, H, G,,, G, H Slide 123 / re lines and coplanar on the cube below? Slide 124 / Plane and plane G intersect at? Yes No line Line G they don't intersect Slide 125 / Planes, G, and G intersect at? line G point point line Slide 126 / Name another point that is in the same plane as points, G, and H. G H

22 Slide 127 / Name a point that is coplanar with points,, and. Slide 128 / Intersecting lines are coplanar. G H lways Sometimes Never Slide 129 / Two planes intersect at exactly one point. lways Sometimes Never Slide 130 / plane can be drawn so that any three points are coplaner lways Sometimes Never Slide 131 / plane containing two points of a line contains the entire line. lways Sometimes Never Slide 132 / our points are noncoplanar. lways Sometimes Never

23 Slide 133 / 206 Slide 134 / Two lines meet at more than one point. lways Sometimes Never ongruence, istance and Length Return to Table of ontents Slide 135 / 206 ongruence 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 Slide 136 / 206 ongruence y 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. which is read as "a is congruent to b." Slide 137 / 206 ongruence 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. Slide 138 / 206 nd 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. ongruence a b a b

24 Slide 139 / 206 ongruence Would the same be true for any two rays? Slide 140 / 206 ongruence gain, all rays are infinitely long, so they have the same length. nd once their vertices and any other point on both rays overlap, all of their points will overlap. ll rays are congruent. a b a b Slide 141 / 206 ongruence Slide 142 / 206 ongruence Would the same be true of all line segments? 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. a b a b Slide 143 / 206 istance and Length Slide 144 / 206 istance and Length While distance and length are related terms, they are also different. t your table, come up with definitions of istance and Length which show how they are related and how they are different. istance: Length: istance 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.

25 Slide 145 / 206 istance and Length Ruler Postulate: ny 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. Slide 146 / 206 istance and Length or instance, we can indicate that on the below number line: Point is located at the position of 0. Point is located at Slide 147 / 206 Slide 148 / 206 istance and Length We can say that points and are 7 apart since we have to move 7 units of measure to get from the location at 0 to that at +7. lso, we can construct line segment and note that it has a length of 7. istance and Length ny line segment which has a length of 7 will be congruent with, even if it needs to be rotated or moved to overlap it. ll such segments have the same length regardless of orientation. So, segment and are congruent and have length 7. So, two points which are 7 apart can be connected by a line segment of length Slide 149 / 206 Slide 150 / 206 istance and Length istance and Length What is the distance of the line below? Is that answer positive or negative? ll 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

26 Slide 151 / 206 istance Slide 152 / What is the location of point? 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. istance is always a positive number. o you remember a term we use in physics to describe a distance which has a direction and could have a negative value? Slide 153 / What is the location of point? Slide 154 / What is the distance from to? Slide 155 / What is the distance from to? Slide 156 / What is the distance from to?

27 Slide 157 / 206 alculating istance Sometimes it is easier to calculate the distance between two points rather than count the steps between them. irst, 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 158 / 206 alculating istance Let's calculate the distance between and. irst, note that is at -7 and is at 0 Then, subtract those numbers: -7 - (0) = -7 [lways 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 and is Slide 159 / 206 alculating istance Let's do the same calculation, but this time let's reverse how we do the subtraction, let's subtract from. Slide 160 / What's the distance between and? irst, let's note that is at -7 and 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 and is 7, calculated either way Slide 161 / 206 Slide 162 / What's the distance between two points if one is located at +125 and the other is located at -350? 55 What's the distance between two points if one is located at -540 and the other is located at -180?

28 Slide 163 / 206 xample ind the measure of each segment in centimeters. Slide 164 / ind a segment that is 4 cm long. cm a. b. = 8-2 = 6 cm = 1.5 cm cm Slide 165 / 206 Slide 166 / ind a segment that is 6.5 cm long. 58 ind a segment that is 3.5 cm long. cm cm Slide 167 / ind a segment that is 2 cm long. Slide 168 / ind a segment that is 5.5 cm long. cm cm

29 Slide 169 / If point was placed at 3.5 cm on the ruler, how far from point would it be? 5 cm 4 cm 3.5 cm 4.5 cm Slide 170 / 206 Segment ddition 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. cm Slide 171 / 206 dding Line Segments If is between and, then + =. Slide 172 / 206 dding Line Segments This works for any number of segments on a line. lternatively If + =, then is between and = Slide 173 / 206 xample Label the line and find x given that: P lies between K and M on a line. Slide 174 / 206 xample Given: = 27 = = 5 = 6 PM= 2x + 4 MK= 14x - 56 PK = x + 17 ind:

30 Slide 175 / 206 xample P,, L, and M are collinear and are in the following order: a) P is between and M Slide 176 / What is the length of Segment? b) L is between M and P raw a diagram and solve for x, given: ML = 3x +16 PL = 2x +11 M = 3x +140 Hint: always start these problems by placing the information you have into the diagram. P = 3x + 13 Slide 177 / What is the length of Segment? Slide 178 / What is the length of Segment? Slide 179 / What is the length of Segment? Slide 180 / What is the length of Segment?

31 Slide 181 / What is the length of Segment? Slide 182 / What is the length of Segment? Slide 183 / X,, and Y are collinear points, with Y between and X. Place the points on the line and solve for x, given: X = 6x XY = 15x - 7 Y = x - 12 Slide 184 / Q, X, and R are collinear points, with X between R and Q. raw a diagram and solve for x, given: XQ = 15x + 10 RQ = 2x XR = 7x +1 R X Q X Y Slide 185 / 206 Slide 186 / , K, and V are collinear points, with K between V and. raw a diagram and solve for x, given: K = 5x V = 15x KV = 4x +149 onstructions and Loci K V Return to Table of ontents

32 Slide 187 / 206 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. Slide 188 / 206 The ircle 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. That can be done algebraically, but it can also be done with the use of drawing equipment such as a straight edge and compass. 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 / 206 uclid and ircles Slide 190 / 206 uclid and ircles Third Postulate: To describe a circle with any center and distance. efinition 15: 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. This postulate says that we can draw a circle of any radius, placing its center where we choose. The straight lines referenced here are the radii which are of equal length from the center to the points on the circle Slide 191 / 206 uclid and ircles Slide 192 / 206 Introduction to onstructions efinition 16: nd 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. In addition to a pencil, we will be using two tools to construct geometric figures a straight edge and a compass. straight edge allows us to draw a straight line, which we are allowed to do between any two points. compass allows us to draw a circle. Try the compass to the right. You can use the pencil to rotate the compass

33 Slide 193 / 206 Introduction to onstructions The sharp point of a compass is placed at the center of the circle. The pencil then draws the circle. or 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 / 206 Try this! 1) reate a circle using the segment below. M r center circle Slide 195 / 206 Try this! Slide 196 / 206 onstructing ongruent Segments 2) reate a circle using the segment below. Let's use these tools to create a line segment which is congruent with the given line segment. H We will first do this with a straight edge and compass. G M Slide 197 / 206 onstructing ongruent Segments irst, use your straight edge to draw a line which is longer than and includes Point, such as Line a below. Slide 198 / 206 onstructing ongruent Segments Then, stretch your compass between points and. a a

34 Slide 199 / 206 onstructing ongruent Segments The compass can now be used to draw an arc with any center with the radius of, how do you think we could use that to create a congruent segment on Line a with as an endpoint? Slide 200 / 206 onstructing ongruent Segments Then, keeping the compass unchanged, place its point at and make an arc through line a. ll the points on that arc are a distance from. The point where the arc intersects the line, is that distance from and on the line. a a Slide 201 / 206 onstructing ongruent Segments Then, draw Point at the intersection of the arc and line a. Point is on the line at a distance of from. Slide 202 / 206 onstructing ongruent Segments Segment is congruent with segment, which was our objective. a a Slide 203 / 206 Slide 204 / 206 Try this! Try this! 3)onstruct a congruent segment on the given line. 4) onstruct a congruent segment on the given line. N M J K L I

35 Slide 205 / 206 Slide 206 / 206 ynamic Geometric Software lick on the image below to watch a video demonstrating constructing congruent segments using ynamic Geometric Software