Points, Lines, Planes, and ngles Points, Lines, Planes, & ngles www.njctl.org Table of ontents Points, Lines, & Planes Line Segments Simplifying Perfect Square Radical Expressions Rational & Irrational Numbers Simplifying Non Perfect Square Radicands Pythagorean Theorem istance between points Midpoint formula ngles & ngle Relationships ngle ddition Postulate efinitions Points, Lines, & Planes n "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry: Points point has no dimensions (length, width, height), it is usually represent by a capital letter and a dot on a page. It shows position only. Lines composed of an unlimited number of points along a straight path. line has no width or height and extends infinitely in opposite directions. Return to Table of ontents Planes a flat surface that extends indefinitely in twodimensions. plane has no thickness. 1
Points, Lines, Planes, and ngles Points & Lines television picture is composed of many dots placed closely together. ut if you look very closely, you will see the spaces.... Points are labeled with letters. (Points,, or ) Lines are named by using any two points OR by using a single lower cased letter. rrowheads show the line continues without end in opposite directions. Line,, or all refer to the same line However, in geometry, a line is composed of an unlimited/infinite number of points. There are no spaces between the point that make a line. You can always find a point between any two other points. The line above would b called line or line Line a ollinear Points Points, E, and F above are called collinear points, meaning they all lie on the same line. Points,, and are NOT collinear point since they do not lie on the same (one) line. Example Give six different names for the line that contains points U, V, and W. Postulate: ny two points are always collinear. Line,, or all refer to the same line Line a (click) 2
Points, Lines, Planes, and ngles Postulate: two lines intersect at exactly one point. If two non parallel lines intersect in a plane they do so at only one point. Example a. Name three points that are collinear b. Name three sets of points that are noncollinear c. What is the intersection of the two lines? and intersect at K. a.,, b.,, /,, /,, (others) Move Rays are also portions of a line. or Suppose point is between points and is read ray. Rays start at an initial point, here endpoint, and continues infinitely in one direction. Rays and are opposite rays. Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line. Recall: Since,, and all lie on the same line, we know they are collinear points. Ray has a different initial point, endpoint, and continues infinitely in the direction marked. Rays and are NOT the same. They have different initial points and extend in different directions. Similarly, segments and rays are called collinear, if they lie on the same line. Segments, rays, and lines are also called coplanar if they all lie on the same plane. 3
Points, Lines, Planes, and ngles Example Name a point that is collinear with the given points. Example Name two opposite rays on the given line a. R and P b. M and Q c. S and N d. O and P e. f. g. h. 1 is the same as. True False 2 is the same as. True False Hint Read the notation carefully. re they asking about lines, line Move segments, or rays? 4
Points, Lines, Planes, and ngles 3 Line p contains just three points. True False 4 Points, H, and E are collinear. True False Hint Remember that even though only three points are marked, a line is composed of an infinite number of points. You Move can always find another point in between two other points. 5Ray LJ and ray JL are opposite rays. Explain your answer. Yes No 6Which of the following are opposite rays? and and and and 5
Points, Lines, Planes, and ngles 7Name the initial point of re the three points collinear? If they are, name the line they lie on. J K L a. L, K, J b. N, I, M c. M, N, K d. P, M, I Planes ollinear points are points that are on the same line. F,G, and H are three collinear points. J,G, and K are three collinear points. J,G, and H are three non collinear points. F, G, H, and I are coplanar. F, G, H, and J are also coplanar, but the plane is not drawn. oplanar points are points that lie on the same plane. F,G, and H are coplanar in addition to being collinear. G, I, and K are non coplanar and non collinear. ny three noncollinear points can name a plane. Planes can be named by any three noncollinear points: plane KMN, plane LKM, or plane KNL or, by a single letter such as Plane R (all name the same plane) oplanar points are points that lie on the same plane: Points K, M, and L are coplanar Points O, K, and L are non coplanar in the diagram above However, you could draw a plane to contain any three points 6
Points, Lines, Planes, and ngles Postulate: If two planes intersect, they intersect along exactly one line. Postulate: Through any three noncollinear points there is exactly one plane. The intersection of the two planes above is shown by line s another example, picture the intersections of the four walls in a room with the ceiling or the floor. You can imagine a line laying along the intersections of these planes. Example Name the following points: point not in plane HIE 8Line does not contain point R. re points R,, and collinear? Yes No point not in plane GIE Two points in both planes Two points not on 7
Points, Lines, Planes, and ngles 9Plane LMN does not contain point P. re points P, M, and N coplanar? Yes No 10Plane QRS contains. re points Q, R, S, and V coplanar? (raw a picture) Yes No Hint: What do we know about any three points? Move 11Plane JKL does not contain. re points J, K, L, and N coplanar? Yes No 12 and intersect at Point Point Point Point 8
Points, Lines, Planes, and ngles 13Which group of points are noncoplanar with points,, and F on the cube below. E, F,,,, G, E, H, G, F, E, G, H 14re lines and coplanar on the cube below? Yes No 15Plane and plane G intersect at? line Line G they don't intersect 16Planes, G, and EG intersect at? line point point line 9
Points, Lines, Planes, and ngles 17Name another point that is in the same plane as points E, G, and H F 18Name a point that is coplanar with points E, F, and H 19Intersecting lines are coplanar. lways Sometimes Never 20Two planes intersect at exactly one point. lways Sometimes Never 10
Points, Lines, Planes, and ngles 21 plane can be drawn so that any three points are coplaner lways Sometimes Never 22 plane containing two points of a line contains the entire line. lways Sometimes Never 23Four points are noncoplanar. 24Two lines meet at more than one point. lways lways Sometimes Sometimes Never Never Look what happens if I place line y directly on top of line x. Hint 11
Points, Lines, Planes, and ngles Line Segments or Line Segments Line segments are portions of a line. or endpoint is read segment. endpoint Return to Table of ontents Line Segment or are different names for the same segment. It consists of the endpoints and and all the points on the line between them. Ruler Postulate On a number line, every point can be paired with a number and every number can be paired with a point. oordinates indicate the point's position on the number line. The symbol F stands for the length of. This distance from to F can be found by subtracting the two coordinates and taking the absolute value. E F Why did we take the bsolute Value when calculating distance? In our previous slide, we were seeking the distance between two points. istance is a physical quantity that can be measured distances cannot be negative. 10 9 8 7 6 coordinate 5 4 3 2 1 istance F = 8 6 = 14 0 1 2 3 4 5 6 7 8 9 10 F coordinate When you take the absolute value between two numbers, the order in which you subtract the two numbers does not matter 12
Points, Lines, Planes, and ngles efinition: ongruence Equal in size and shape. Two objects are congruent if they have the same dimensions and shape. Roughly, 'congruent' means 'equal', but it has a precise meaning that you should understand completely when you consider complex shapes. Line Segments are congruent if they have the same length. ongruent lines can be at any angle or orientation on the plane; they do not need to be parallel. Read as: "The line segment E is congruent to line segment HI." efinition: Parallel Lines Lines are parallel if they lie in the same plane, and are the same distance apart over their entire length. That is, they do not intersect. Example Find the measure of each segment in centimeters. 25Find a segment that is 4 cm long cm a. b. = = cm 13
Points, Lines, Planes, and ngles 26Find a segment that is 3.5 cm long 27Find a segment that is 2 cm long cm cm 28If point F was placed at 3.5 cm on the ruler, how far from point E would it be? Segment ddition Postulate 5 cm 4 cm 3.5 cm 4.5 cm cm Simply said, if you take one part of a segment (), and add it to another part of the segment (), you get the entire segment. The whole is equal to the sum of its parts. 14
Points, Lines, Planes, and ngles Example The segment addition postulate works for three or more segments if all the segments lie on the same line (i.e. all the points are collinear). Start by filling in the information you are given E E In the diagram, E = 27, =, E = 5, and = 6 In the diagram, E = 27, =, E = 5, and = 6 27 Find and E 6 5 an you finish the rest? = E = K, M, and P are collinear with P between K and M. PM = 2x+4, MK = 14x 56, and PK = x+17 Solve for x. Example P,, L, and M are collinear and are in the following order: a) P is between and M b) L is between M and P raw a diagram and solve for x, given: ML = 3x +16, PL = 2x +11, M = 3x +140, and P = 3x + 13 1) First, arrange the points in order and draw a diagram a) PM b) PLM 2) Segment addition postulate gives 3x+13 + 2x+11 + 3x+16 = 3x+140 3) ombine like terms and isolate/solve for the variable x 8x + 40 = 3x + 140 5x + 40 = 140 5x = 100 x = 20 15
Points, Lines, Planes, and ngles 29We are given the following information about the collinear points: 30We are given the following information about the collinear points: What is,, and? What is? 31We are given the following information about the collinear points: 32We are given the following information about the collinear points: What is? What is? 16
Points, Lines, Planes, and ngles 33We are given the following information about the collinear points: 34We are given the following information about the collinear points: What is? What is? 35X,, and Y are collinear points, with Y between and X. raw a diagram and solve for x, given: X = 6x + 151 XY = 15x 7 Y = x 12 36Q, 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 + 131 XR = 7x +1 17
Points, Lines, Planes, and ngles 37, K, and V are collinear points, with K between V and. raw a diagram and solve for x, given: K = 5x V = 15x + 125 KV = 4x +149 Simplifying Perfect Square Radical Expressions Return to Table of ontents an you recall the perfect squares from 1 to 169? 1 2 = 8 2 = 2 2 = 9 2 = 3 2 = 10 2 = 4 2 = 11 2 = Square Root Of Number Recall: If b 2 = a, then b is a square root of a. Example: If 4 2 = 16, then 4 is a square root of 16 What is a square root of 25? 64? 100? 5 2 = 12 2 = 6 2 = 13 2 = 20 2 = 7 2 = 18
Points, Lines, Planes, and ngles Square Root Of Number Square roots are written with a radical symbol Positive square root: = 4 Negative square root: = 4 Positive & negative square roots: = 4 Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal 16. Is there a difference between Which expression has no real roots? Evaluate the expression &? Evaluate the expression 38? is not real 19
Points, Lines, Planes, and ngles 39 =? 40 41 =? 3 3 No real roots Rational & Irrational Numbers Return to Table of ontents 20
Points, Lines, Planes, and ngles Rational & Irrational Numbers is rational because the radicand (number under the radical) is a perfect square If a radicand is not a perfect square, the root is said to be irrational. 42Rational or Irrational? Rational Irrational Ex: 43Rational or Irrational? 44Rational or Irrational? Rational Irrational Rational Irrational 21
Points, Lines, Planes, and ngles What happens when the radicand is not a perfect square? Simplifying Non Perfect Square Radicands Rewrite the radicand as a product of its largest perfect square factor. Simplify the square root of the perfect square. Return to Table of ontents When simplified form still contains a radical, it is said to be irrational. Try These. Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. Ex: Not simplified! Keep going! Finding the largest perfect square factor results in less work: Note that the answers are the same for both solution processes 22
Points, Lines, Planes, and ngles 45Simplify 46Simplify already in simplified form already in simplified form 47Simplify 48Simplify already in simplified form already in simplified form 23
Points, Lines, Planes, and ngles 49Simplify 50Simplify already in simplified form already in simplified form 51Which of the following does not have an irrational simplified form? 2 24
Points, Lines, Planes, and ngles 52Simplify 53Simplify 54Simplify 55Simplify 25
Points, Lines, Planes, and ngles 56Simplify The Pythagorean Theorem Return to Table of ontents Pythagorean Theorem Pythagoras was a philosopher, theologian, scientist and mathematician born on the island of Samos in ancient Greece and lived from c. 570 c. 495. Using the Pythagorean Theorem In the Pythagorean Theorem, c always stands for the longest side. In a right triangle, the longest side is called the hypotenuse. The hypotenuse is the side opposite the right angle. The Pythagorean Theorem c 2 = a 2 + b 2 states that in a right triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides. c a b lick to see a Visual Proof Proof Proof c 2 = a 2 + b 2 5 a =? 3 25 = a 2 + 9 9 9 a 2 16 = = a 4 = a You will use the Pythagorean Theorem often. 26
Points, Lines, Planes, and ngles Example 57What is the length of side c? Hint: The longest side of a triangle is called Move the? 58What is the length of side a? 59What is the length of c? Hint: lways determine which side is the hypotenuse first Move 27
Points, Lines, Planes, and ngles 60What is the length of the missing side? 61What is the length of side b? 62What is the measure of x? 63 alculate the value of the missing side. Leave your answer in simplest radical form. x 8 17 8 10 28
Points, Lines, Planes, and ngles 64 alculate the value of the missing side. Leave your answer in simplest radical form. 18 65 alculate the value of x. Leave your answer in simplest radical form. 18 6 12 Pythagorean Triples are three positive integers for side lengths that satisfy a 2 + b 2 = c 2 66 triangle has sides 30, 40, and 50, is it a right triangle? Yes No ( 3, 4, 5 ) ( 5, 12, 13) (6, 8, 10)( 7, 24, 25) ( 8, 15, 17) ( 9, 40, 41) (10, 24, 26) (11, 60, 61) (12, 35, 37) (13, 84, 85) etc. There are many more. Remembering some of these combinations may save you some time 29
Points, Lines, Planes, and ngles 67 triangle has sides 9, 12, and 15, is it a right triangle? 68 triangle has sides 3, 2, and 5, is it a right triangle? Yes No Yes No istance omputing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles. omputing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle. Return to Table of ontents 30
Points, Lines, Planes, and ngles Relationship between the Pythagorean Theorem & istance Formula The Pythagorean Theorem states a relationship among the sides of a right triangle. c 2 = a 2 + b 2 c a The distance formula calculates the distance using the points' coordinates. (x2, y2) c istance The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula. The istance Formula The distance 'd' between any two points with coordinates (x 1, y 1) and (x 2, y 2) is given by the formula: b (x1, y1) (x2, y1) d = The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side. Note: recall that all coordinates are (x coordinate, y coordinate). Example alculate the distance from Point K to Point I (x 1, y 1) (x 2, y 2) 69alculate the distance from Point J to Point K Label the points it does not matter which one you label point 1 and point 2. Your answer will be the same. Plug the coordinates into the distance formula d = KI = KI = = = 31
Points, Lines, Planes, and ngles 70alculate the distance from H to K 71alculate the distance from Point G to Point K 72alculate the distance from Point I to Point H 73alculate the distance from Point G to Point H 32
Points, Lines, Planes, and ngles 74 alculate the distance between ( 3, 7) and (8, 2). 75 alculate the distance between (3, 7) and (1, 3). 76 alculate the distance between (3, 0) and ( 10, 6). Midpoints Return to Table of ontents 33
Points, Lines, Planes, and ngles Midpoint of a line segment number line can help you find the midpoint of a segment. The midpoint of GH, marked by point M, is 1. Midpoint Formula Theorem The midpoint of a segment joining points with coordinates (x 1, y 1) and (x 2, y 2) is the point with coordinates Here's how you calculate it using the endpoint coordinates. Take the coordinates of the endpoint G and H, add them together, and divide by two. = = 1 alculating Midpoints in a artesian Plane Segment PQ contain the points (2, 4) and (10, 6). The midpoint M of is the point halfway between P and Q. Just as before, we find the average of the coordinates. 77Find the midpoint coordinates (x,y) of the segment connecting points (1,2) and (5,6) (4, 3) (3, 4) (6, 8) (2.5, 3) Remember that points are written with the x coordinate first. (x, y) Hint: (, ) The coordinates of M, the midpoint of PQ, are (6, 5) lways label the points coordinates first 34
Points, Lines, Planes, and ngles 78Find the midpoint coordinates (x,y) of the segment connecting the points ( 2,5) and (4, 3) ( 1, 1) ( 3, 8) ( 8, 3) (1, 1) 79Find the coordinates of the midpoint (x, y) of the segment with endpoints R( 4, 6) and Q(2, 8) ( 1, 1) (1, 1) ( 1, 1) (1, 1) 80Find the coordinates (x, y) of the midpoint of the segment with endpoints ( 1, 3) and ( 7, 9) ( 3, 3) (6, 4) ( 4, 6) (4, 6) 81Find the midpoint (x, y) of the line segment between ( 1, 3) and (2,2) (3/2, 5/2) (1/2, 5/2) (1/2, 3) (3, 1/2) 35
Points, Lines, Planes, and ngles Example: Finding the coordinates of an endpoint of an segment 82Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0) ( 1, 2) ( 2, 1) (4, 2) (2, 4) Use the midpoint formula to write equations using x and y. 83Find the other endpoint of the segment with the endpoint (1, 4) and midpoint (5, 2) (11, 8) (9, 0) (9, 8) (3, 1) ngles & ngle Relationships Return to Table of ontents 36
Points, Lines, Planes, and ngles Identifying ngles n angle is formed by two rays with a common endpoint (vertex) The angle shown can be called,, or. When there is no chance of confusion, the angle may also be identified by its vertex. The sides of are and (Vertex) 32 (Side) (Side) The measure of the angle is 32 degrees. "The measure of is equal to the measure of..." Two angles that have the same measure are congruent angles. Exterior Interior We read this as The single mark through the arc shows that the angle measures are equal is congruent to The area between the rays that form an angle is called the interior. The exterior is the area outside the angle. ngle Measures ngles are measured in degrees, using a protractor. Every angle has a measure from 0 to 180 degrees. ngles can be drawn any size, the measure would still be the same. Example K L M N is a 23 degree angle The measure of is 23 degrees is a 119 degree angle The measure of is 119 degrees P hallenge Questions J O In and, notice that the vertex is written in between the sides 37
Points, Lines, Planes, and ngles ngle Relationships Once we know the measurements of angles, we can categorize them into several groups of angles: 0 < acute < 90 90 < obtuse < 180 omplementary ngles pair of angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the angles is said to be the complement of the other. These two angles are complementary (58 + 32 = 90 ) right = 90 Two lines or line segments that meet at a right angle are said to be perpendicular. straight = 180 180 180 < reflex angle < 360 Link We can rearrange the angles so they are adjacent, i.e. share a common side and a vertex. omplementary angles do not have to be adjacent. If two adjacent angles are complementary, they form a right angle Supplementary ngles Supplementary angles are pairs of angles whose measurements sum to 180 degrees. Supplementary angles do not have to be adjacent or on the same line; they can be separated in space. One angle is said to be the supplement of the other. efinition: djacent ngles are angles that have a common ray coming out of the vertex going between two other rays. In other words, they are angles that are side by side, or adjacent. If the two supplementary angles are adjacent, having a common vertex and sharing one side, their non shared sides form a line. linear pair of angles are two adjacent angles whose noncommon sides on the same line. line could also be called a straight angle with 180 38
Points, Lines, Planes, and ngles Example Solution: hoose a variable for the angle I'll choose "x" Example Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles? Let x = the angle Since the angles are complementary we know their sum must equal 90 degrees. 90 = 2x + x 90 = 3x 30 = x 84n angle is 34 more than its complement. What is its measure? 85n angle is 14 less than its complement. What is the angle's measure? Hint: hoose a variable for the angle. What is a complement? Move Hint: What is a complement? hoose a variable for the angle Move 39
Points, Lines, Planes, and ngles 86n angle is 98 more than its supplement. What is the measure of the angle? 87n angle is 74 less than its supplement. What is the angle? Hint: hoose a variable for the angle What is a supplement? Move 89 and are a linear pair. What is the value of x if and. 88n angle is 26 more than its supplement. What is the angle? 40
Points, Lines, Planes, and ngles ngle ddition Postulate ngle ddition Postulate if a point S lies in the interior of PQR, then PQS + SQR = PQR. 32 26 58 Return to Table of ontents m PQS = 32 + m SQR = 26 m PQR = 58 Just as from the Segment ddition Postulate, "The whole is the sum of the parts" Example 90 Given m = 22 and m = 46. Find m Hint: lways label your diagram with the information given Move 41
Points, Lines, Planes, and ngles 91 Given m OLM = 64 and m OLN = 53. Find m NLM 92 Given m = 95 and m = 48. 28 15 11 117 64 53 Find m 93 Given m KLJ = 145 and m KLH = 61. Find m HLJ 94 Given m TRQ = 61 and m SRQ = 153. Find m SRT 42
Points, Lines, Planes, and ngles 95 is in the interior of TUV. 96 is in the interior of. If m TUV = (10x + 72)⁰, If m = (11x + 66)⁰, m TU = (14x + 18)⁰ and m = (5x + 3)⁰ and m UV = (9x + 2)⁰ m = (13x + 7)⁰ solve for x. solve for x. Hint: raw a diagram and label it with the given information Move Hint: raw a diagram and label it with the given information Move 97 F is in the interior of QP. If m QP = (3x + 44)⁰, m FQP = (8x + 3)⁰ and m QF= (5x + 1)⁰ solve for x 43