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Points, Lines, Planes, and Angles Points, Lines, Planes, & Angles Table of Contents Points, Lines, & Planes Line Segments Simplifying Perfect Square Radical Expressions Rational & Irrational Numbers Simplifying NonPerfect Square Radicands Pythagorean Theorem Distance between points Midpoint formula Angles & Angle Relationships Angle Addition Postulate www.njctl.org Definitions An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry: Points, Lines, & Planes Points A 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. A line has no width or height and extends infinitely in opposite directions. Return to Table of Contents Planes a flat surface that extends indefinitely in two dimensions. A plane has no thickness. 1 Points, Lines, Planes, and Angles Points are labeled with letters. (Points A, B, or C) Points & Lines A television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces. Lines are named by using any two points OR by using a single lowercased letter. Arrowheads show the line continues without end in opposite directions. Line , , or all refer to the same line .................................... B A 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. Line a The line above would b called line or line Collinear Points Points D, E, and F above are called collinear points, meaning they all lie on the same line. Example Points A, B, and C are NOT collinear point since they do not lie on the same (one) line. Give six different names for the line that contains points U, V, and W. Postulate: Any two points are always collinear. Line , , or all refer to the same line Answer (click) Line a 2 Points, Lines, Planes, and Angles Postulate: two lines intersect at exactly one point. If two nonparallel 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? Answer and intersect at K. Rays are also portions of a line. a. A, D, C b. A,B,D / A,C,B / C,D,B (others) Move Suppose point C is between points A and B or Rays and are opposite rays. is read ray AB. Rays start at an initial point, here endpoint A, and continues infinitely in one direction. Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line. Recall: Since A, B, and C all lie on the same line, we know they are collinear points. Ray has a different initial point, endpoint B, and continues infinitely in the direction marked. 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. Rays and are NOT the same. They have different initial points and extend in different directions. 3 Points, Lines, Planes, and Angles Example Example Name a point that is collinear with the given points. Name two opposite rays on the given line e. b. M and Q f. c. S and N g. d. O and P h. is the same as . True True False False Answer 2 is the same as . Answer 1 a. R and P Hint Read the notation carefully. Are they asking about lines, line Move segments, or rays? 4 Points, Lines, Planes, and Angles Line p contains just three points. 4 Points D, H, and E are collinear. True True False False Answer Answer 3 Hint Remember that even though only three points are marked, a line is composed of an infinite number of points. You can Move always find another point in between two other points. 5Ray LJ and ray JL are opposite rays. 6Which of the following are opposite rays? Explain your answer. Yes A and No B and C and No, Opposite Rays have same endpoint but point in opposite directions Answer Answer D and 5 Points, Lines, Planes, and Angles Are the three points collinear? If they are, name the line they lie on. 7Name the initial point of A J a. L, K, J b. N, I, M c. M, N, K d. P, M, I B K Answer C L Planes Collinear 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 noncollinear points. F, G, H, and I are coplanar. F, G, H, and J are also coplanar, but the plane is not drawn. Coplanar 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 noncoplanar and noncollinear. Any 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) Coplanar points are points that lie on the same plane: Points K, M, and L are coplanar Points O, K, and L are noncoplanar in the diagram above However, you could draw a plane to contain any three points 6 Points, Lines, Planes, and Angles Postulate: Through any three noncollinear points there is exactly one plane. A B Postulate: If two planes intersect, they intersect along exactly one line. The intersection of the two planes above is shown by line As 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 8Line BC does not contain point R. Are points R, B, and C collinear? Name the following points: Yes A point not in plane HIE No Two points in both planes Answer A point not in plane GIE Two points not on 7 Points, Lines, Planes, and Angles 9Plane LMN does not contain point P. Are points P, M, and N coplanar? 10Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture) Yes No No Answer Answer Yes Hint: Move 11Plane JKL does not contain . Are points J, K, L, and N coplanar? 12 and intersect at A Yes Answer No Point A B Point B C Point C D Point D Answer What do we know about any three points? 8 Points, Lines, Planes, and Angles 13Which group of points are noncoplanar with points A, B, and F on the cube below. 14Are lines and coplanar on the cube below? A E, F, B, A B A, C, G, E Yes C D, H, G, C No Answer Answer D F, E, G, H 15Plane ABC and plane DCG intersect at _____? A C 16Planes ABC, GCD, and EGC intersect at _____? A line B line DC B point C C Line CG C point A D they don't intersect B Answer Answer D line B 9 Points, Lines, Planes, and Angles 17Name another point that is in the same plane as 18Name a point that is coplanar with points E, F, and C points E, G, and H A H A B B B B C C D C D D A 19Intersecting lines are __________ coplanar. Answer Answer D F C D 20Two planes ____________ intersect at exactly one point. A Always A Always B Sometimes B Sometimes C Never Answer Answer C Never 10 Points, Lines, Planes, and Angles 21A plane can __________ be drawn so that any three points are coplaner 22A plane containing two points of a line __________ contains the entire line. A Always A Always B Sometimes B Sometimes C Never Answer Answer C Never A Always B Sometimes B Sometimes C Never C Never Answer A Always Answer 24Two lines ________________ meet at more than one point. 23Four points are ____________ noncoplanar. Look what happens if I place line y directly on top of line x. Hint 11 Points, Lines, Planes, and Angles Line Segments or Line segments are portions of a line. or Line Segments endpoint endpoint is read segment AB. Return to Table of Contents Line Segment or are different names for the same segment. It consists of the endpoints A and B and all the points on the line between them. Ruler Postulate Why did we take the Absolute Value when calculating distance? On a number line, every point can be paired with a number and every number can be paired with a point. Coordinates indicate the point's position on the number line. The symbol AF stands for the length of . This distance from A to F can be found by subtracting the two coordinates and taking the absolute 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 F coordinate A coordinate 9 10 In our previous slide, we were seeking the distance between two points. Distance is a physical quantity that can be measured distances cannot be negative. When you take the absolute value between two numbers, the order in which you subtract the two numbers does not matter Distance AF = |8 6| = 14 12 Points, Lines, Planes, and Angles Definition: Congruence 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. Congruent lines can be at any angle or orientation on the plane; they do not need to be parallel. Read as: "The line segment DE is congruent to line segment HI." Definition: 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 25Find a segment that is 4 cm long A B C Answer Find the measure of each segment in centimeters. D cm a. b. = cm = 13 Points, Lines, Planes, and Angles 27Find a segment that is 2 cm long 26Find a segment that is 3.5 cm long C B Answer B Answer A A C D D cm cm 28If point F was placed at 3.5 cm on the ruler, how far from point E would it be? Segment Addition Postulate B 4 cm C 3.5 cm Answer A 5 cm D 4.5 cm AB BC AC cm Simply said, if you take one part of a segment (AB), and add it to another part of the segment (BC), you get the entire segment. The whole is equal to the sum of its parts. 14 Points, Lines, Planes, and Angles Example Start by filling in the information you are given AE 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). AB CD BC DE In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 27 Find CD and BE || Can you finish the rest? K, M, and P are collinear with P between K and M. PM = 2x+4, MK = 14x56, and PK = x+17 Solve for x. || 6 5 CD = BE = 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, and PB = 3x + 13 1) First, arrange the points in order and draw a diagram a) BPM b) BPLM 2) Segment addition postulate gives 3x+13 + 2x+11 + 3x+16 = 3x+140 3) Combine 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 Angles 29We are given the following information about the collinear points: 30We are given the following information about the collinear points: What is ? Answer Answer What is , , and ? 32We are given the following information about the collinear points: 31We are given the following information about the collinear points: Answer Answer What is ? What is ? 16 Points, Lines, Planes, and Angles 33We are given the following information about the collinear points: 34We are given the following information about the collinear points: What is ? 35X, B, and Y are collinear points, with Y between B and X. Draw a diagram and solve for x, given: BX = 6x + 151 XY = 15x 7 BY = x 12 Answer Answer What is ? 36Q, 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 Answer Answer 7x + 1 + 15x + 1 22x + 11 = 2 20x = 1 x = 6 17 Points, Lines, Planes, and Angles 37B, 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 Simplifying Perfect Square Radical Expressions Return to Table of Contents Answer 4x + 149 + 5x = 15x + 125 9x + 149 = 15x + 125 6x = 24 x = 4 Can you recall the perfect squares from 1 to 169? 12 = 82 = 2 2 = 9 = 2 3 = 10 = 2 4 = 11 = 5 = 122 = 62 = 132 = 202 = 2 2 2 Square Root Of A Number Recall: If b 2 = a, then b is a square root of a. Example: If 42 = 16, then 4 is a square root of 16 What is a square root of 25? 64? 100? 2 72 = 18 Points, Lines, Planes, and Angles Is there a difference between Square Root Of A Number Square roots are written with a radical symbol & ? Positive square root: = 4 Which expression has no real roots? Negative square root: = 4 Positive & negative square roots: = 4 Evaluate the expression Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal 16. Evaluate the expression 38 ? is not real 19 Points, Lines, Planes, and Angles 39 = ? 40 41 = ? A 3 B 3 C No real roots Rational & Irrational Numbers Return to Table of Contents 20 Points, Lines, Planes, and Angles Rational & Irrational Numbers is rational because the radicand (number under the radical) is a perfect square 42Rational or Irrational? A Rational B Irrational B Irrational If a radicand is not a perfect square, the root is said to be irrational. Ex: 43Rational or Irrational? A Rational 44Rational or Irrational? B Irrational A Rational 21 Points, Lines, Planes, and Angles What happens when the radicand is not a perfect square? Rewrite the radicand as a product of its largest perfect square factor. Simplifying NonPerfect Square Radicands Return to Table of Contents Try These. Simplify the square root of the perfect square. When simplified form still contains a radical, it is said to be irrational. 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 Angles 45Simplify 46Simplify A A B B C C D already in simplified form 47Simplify D already in simplified form 48Simplify A A B B C C D already in simplified form D already in simplified form 23 Points, Lines, Planes, and Angles 49Simplify 50Simplify A A B B C C D already in simplified form 51Which of the following does not have an irrational simplified form? A B C D D already in simplified form 2 24 Points, Lines, Planes, and Angles 52Simplify 53Simplify A A B B C D C D 54Simplify 55Simplify A B B C C D D A 25 Points, Lines, Planes, and Angles 56Simplify A B C D The Pythagorean Theorem Return to Table of Contents Pythagorean Theorem 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. Pythagoras was a philosopher, theologian, scientist and mathematician born on the island of Samos in ancient Greece and lived from c. 570–c. 495 BC. The Pythagorean Theorem c2 = a2 + b2 c b c2 = a2 + b2 a 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. Click to see a Visual Proof Proof 5 3 a = ? 25 = a2 + 9 9 9 2 16 = a =a 4 =a Proof You will use the Pythagorean Theorem often. 26 Points, Lines, Planes, and Angles 57What is the length of side c? Answer Example Answer Hint: The longest side of a triangle is called the? Move 59What is the length of c? 58What is the length of side a? Hint: Answer Answer B Always determine which side is the hypotenuse first Move 27 Points, Lines, Planes, and Angles 60What is the length of the missing side? Answer Answer 61What is the length of side b? 63 Calculate the value of the missing side. Leave your answer in simplest radical form. 62What is the measure of x? x 10 8 Answer 17 Answer 8 28 Points, Lines, Planes, and Angles 64 Calculate the value of the missing side. Leave your answer in simplest radical form. 65 Calculate the value of x. Leave your answer in simplest radical form. 18 Pythagorean Triples are three positive integers for side lengths that satisfy a2 + b 2 = c2 ( 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 12 66A triangle has sides 30, 40 , and 50, is it a right triangle? Yes No Answer 6 Answer Answer 18 29 Points, Lines, Planes, and Angles 67 A triangle has sides 9, 12 , and 15, is it a right triangle? 68 A triangle has sides √3, 2 , and √5, is it a right triangle? Yes No No Answer Answer Yes Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles. Distance Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle. Return to Table of Contents 30 Points, Lines, Planes, and Angles Relationship between the Pythagorean Theorem & Distance Formula The Pythagorean Theorem states a relationship among the sides of a right triangle. The distance formula calculates the distance using the points' coordinates. Distance The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula. (x2, y2) The Distance Formula c = a + b 2 2 2 The distance 'd' between any two points with coordinates (x2, y2) and is given by the formula: (x1, y1) c c a (x1, y1) b d = (x2, y1) 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 (xcoordinate, ycoordinate). 69Calculate the distance from Point J to Point K Example A B Calculate the distance from Point K to Point I (x1, y1) C (x2, y2) D Answer Label the points it does not matter which one you label point 1 and point 2. Your answer will be the same. d = Plug the coordinates into the distance formula KI = KI = D = = 31 Points, Lines, Planes, and Angles 71Calculate the distance from Point G to Point K 70Calculate the distance from H to K A B B C C D D D Answer Answer A 72Calculate the distance from Point I to Point H A 73Calculate the distance from Point G to Point H B A C B D C Answer Answer D 32 Points, Lines, Planes, and Angles 74 Calculate the distance between A(3, 7) and B(8, 2). 75 Calculate the distance between A(3, 7) and B(1, 3). 76 Calculate the distance between A(3, 0) and B(10, 6). Midpoints Return to Table of Contents 33 Points, Lines, Planes, and Angles Midpoint of a line segment A number line can help you find the midpoint of a segment. Midpoint Formula Theorem The midpoint of GH, marked by point M, is 1. (x1, y1) The midpoint of a segment joining points with coordinates and is the point with coordinates (x2, y2) 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 Calculating Midpoints in a Cartesian Plane Segment PQ contain the points (2, 4) and (10, 6). The midpoint M of is the point halfway between P and Q. A (4, 3) B (3, 4) Remember that points are written with the xcoordinate first. (x, y) The coordinates of M, the midpoint of PQ, are (6, 5) D (2.5, 3) Answer C (6, 8) Just as before, we find the average of the coordinates. ( , ) 77Find the midpoint coordinates (x,y) of the segment connecting points A(1,2) and B(5,6) Hint: Always label the points coordinates first 34 Points, Lines, Planes, and Angles 79Find the coordinates of the midpoint (x, y) of the segment with endpoints R(4, 6) and Q(2, 8) A (1, 1) B (3, 8) B (1, 1) C (8, 3) C (1, 1) D (1, 1) D (1, 1) Answer A (1, 1) 80Find the coordinates (x, y) of the midpoint of the segment with endpoints B(1, 3) and C(7, 9) B (1/2, 5/2) C (4, 6) C (1/2, 3) D (3, 1/2) Answer A (3/2, 5/2) B (6, 4) Answer 81Find the midpoint (x, y) of the line segment between A(1, 3) and B(2,2) A (3, 3) D (4, 6) Answer 78Find the midpoint coordinates (x,y) of the segment connecting the points A(2,5) and B(4, 3) 35 Points, Lines, Planes, and Angles 82Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0) Example: Finding the coordinates of an endpoint of an segment A (1, 2) B (2, 1) C (4, 2) Answer D (2, 4) Use the midpoint formula to write equations using x and y. A 83Find the other endpoint of the segment with the endpoint (1, 4) and midpoint (5, 2) A (11, 8) B (9, 0) Angles & Angle Relationships C (9, 8) Answer D (3, 1) C Return to Table of Contents 36 Points, Lines, Planes, and Angles Identifying Angles Two angles that have the same measure are congruent angles. An angle is formed by two rays with a common endpoint (vertex) The angle shown can be called , , or . C The sides of are CB and AB The single mark through the arc shows that the angle measures are equal Exterior When there is no chance of confusion, the angle may also be identified by its vertex B. (Side) Interior 32° B (Vertex) A (Side) We read this as is congruent to The measure of the angle is 32 degrees. The area between the rays that form an angle is called the interior. The exterior is the area outside the angle. "The measure of is equal to the measure of ..." Angle Measures L Angles are measured in degrees, using a protractor. Every angle has a measure from 0 to 180 degrees. Angles can be drawn any size, the measure would still be the D same. M Example K N A P B is a 23° degree angle The measure of is 23° degrees J O C is a 119° degree angle The measure of is 119° degrees Challenge Questions In and , notice that the vertex is written in between the sides 37 Points, Lines, Planes, and Angles Angle Relationships Once we know the measurements of angles, we can categorize them into several groups of angles: 0° < acute < 90° 90° < obtuse < 180° Complementary Angles A 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° straight = 180° 180° 180° < reflex angle < 360° Two lines or line segments that meet at a right angle are said to be perpendicular. Link We can rearrange the angles so they are adjacent, i.e. share a common side and a vertex. Complementary angles do not have to be adjacent. If two adjacent angles are complementary, they form a right angle Supplementary Angles 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. Definition: Adjacent Angles 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 nonshared sides form a line. A linear pair of angles are two adjacent angles whose non common sides on the same line. A line could also be called a straight angle with 180° 38 Points, Lines, Planes, and Angles Example Example Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles? Solution: Choose a variable for the angle I'll choose "x" Let x = the angle Since the angles are complementary we know their sum must equal 90 degrees. 90 = 2x + x 90 = 3x 30 = x 85An angle is 14° less than its complement. 84An angle is 34° more than its complement. Choose a variable for the angle. What is a complement? Move Answer Hint: angle = complement + 34 angle = (90 x) + 34 Answer What is the angle's measure? What is its measure? Hint: What is a complement? Choose a variable for the angle Move 39 Points, Lines, Planes, and Angles 87An angle is 74° less than its supplement. 86An angle is 98 more than its supplement. What is the angle? What is the measure of the angle? Answer Answer angle = supp Hint: Choose a variable for the angle Move What is a supplement? 89 and are a linear pair. What is the value of x if and . 88An angle is 26° more than its supplement. What is the angle? Answer angle = supplement + 26 40 Points, Lines, Planes, and Angles Angle Addition Postulate if a point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR. Angle Addition Postulate 32° 58° 26° Return to Table of Contents m∠PQS = 32° + m∠SQR = 26° m∠PQR = 58° Just as from the Segment Addition Postulate, "The whole is the sum of the parts" 90 Given m∠ABC = 22° and m∠DBC = 46°. Find m∠ABD Answer Example Hint: Always label your diagram with the information given Move 41 Points, Lines, Planes, and Angles 91 Given m∠OLM = 64° and m∠OLN = 53°. Find m∠NLM 92 Given m∠ABD = 95° and m∠CBA = 48°. Find m∠DBC A 28 B 15 C 11 64° Answer 53° Answer D 117 94 Given m∠TRQ = 61° and m∠SRQ = 153°. 93 Given m∠KLJ = 145° and m∠KLH = 61°. Answer Answer Find m∠SRT Find m∠HLJ 42 Points, Lines, Planes, and Angles 95 C is in the interior of ∠TUV. 96 D is in the interior of ∠ABC. m∠DBA = (5x + 3)⁰ and m∠CUV = (9x + 2)⁰ m∠CBD= (13x + 7)⁰ solve for x. solve for x. 11x + 66 = 5x + Answer If m∠CBA = (11x + 66)⁰, m∠TUC = (14x + 18)⁰ and Answer If m∠TUV = (10x + 72)⁰, 11x + 66 = 7x = x = Hint: Hint: Draw a diagram and label it with the given information Move Draw a diagram and label it with the given information Move 97 F is in the interior of ∠DQP. If m∠DQP = (3x + 44)⁰, m∠FQP = (8x + 3)⁰ and m∠DQF= (5x + 1)⁰ Answer solve for x 43