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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website www.njctl.org Points, Lines, Planes, & Angles October 4, 2011 www.njctl.org Setting the PowerPoint View Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 17 for an example.) Table of Contents Points, Lines, & Planes Line Segments Pythagorean Theorem Distance between points Midpoint formula Locus & Constructions Angles & Angle Relationships Angle Addition Postulate Angle Bisectors & Constructions Click on the topic to go to that section Points, Lines, & Planes Return to Table of Contents Definitions An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry: 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. Planes - a flat surface that extends indefinitely in twodimensions. A plane has no thickness. Points & Lines A television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces. A B 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 , , or all refer to the same line Line a Points are labeled with letters. (Points A, B, or C) Lines are named by using any two points OR by using a single lower-cased letter. Arrowheads show the line continues without end in opposite directions. Line , , or , all refer to the same line Line a Collinear Points - Points D, E, and F above are called collinear points, meaning they all lie on the same line. Points A, B, and C are NOT collinear point since they do not lie on the same (one) line. Postulate: Any two points are always collinear. Example Give six different names for the line that contains points U, V, and W. Answer (click) Postulate: two lines intersect at exactly one point. If two non-parallel intersect they do so at only one point. and intersect at K. 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 a. A, D, C b. A,B,D / A,C,B / C,D,B (others) Rays are also portions of a line. or is read ray AB. Rays start at an initial point, here endpoint A, and continues infinitely in one direction. Ray has a different initial point, endpoint B, and continues infinitely in the direction marked. Rays and are NOT the same. They have different initial points and extend in different directions. Suppose point C is between points A and B 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 A, B, and C all lie on the same line, we know they are collinear points. 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. Example Name a point that is collinear with the given points. a. R and P b. M and Q c. S and N d. O and P Example Name two opposite rays on the given line e. f. g. h 1 is the same as . True False Hint Read the notation carefully. Are they asking about lines, line segments, or rays? 2 is the same as True False 3 Line p contains just three points True False Answer Remember that even though only three points are marked, a line is composed of an infinite number of points. You can always find another point in between two other points. 4 Points D, H, and E are collinear. True False 5 Points G, D, and H are collinear. True False 6 Ray LJ and ray JL are opposite rays. Yes No Explain your answer? 7 Which of the following are opposite rays? A and B and C and D and 8 Name the initial point of A J B K C L 9 Name the initial point of A J B K C L Are the three points collinear? If they are, name the line they lie on. a L, K, J b N, I, M c M, N, K d P, M, I 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 non-collinear 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 non-coplanar and non-collinear. 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 non-coplanar in the diagram However, you could draw a plane to contain any three points 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. Postulate: Through any three noncollinear points there is exactly one plane. Example Name the following points: A point not in plane HIE A point not in plane GIE Two points in both planes Two points not on 10 Line BC does not contain point R. Are points R, B, and C collinear? Yes No 11 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No Hint: what do we know about any three points? 12 Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture) Yes No 13 Plane JKL does not contain L, and N coplanar? Yes No . Are points J, K, 14 and A D B C C A D B intersect at 15 Which group of points are noncoplanar with points A,B, and F. A E, F, B, A B A, C, G, E C D, H, G, C D F, E, G, H 16 Are lines and coplanar? Yes No Answer 17 Plane ABC and plane DCG intersect at ? A C B line DC C Line CG D they don't intersect Answer 18 Planes ABC, GCD, and EGC intersect at ? A line B point C C point A D line Answer 19 Name another point that is in the same plane as points E, G, and H A B B C C D D F Answer 20 Name a point that is coplanar with points E, F, and C A H B B C D D A Answer 21 Intersecting lines are __________ coplanar. A Always B Sometimes C Never 22 Two planes ____________ intersect at exactly one point. A Always B Sometimes C Never 23 A plane can __________ be drawn so that any three points are coplaner A Always B Sometimes C Never 24 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes C Never 25 Four points are ____________ noncoplanar. A Always B Sometimes C Never 26 Two lines ________________ meet at more than one point. A Always B Sometimes C Never Look what happens if I place line y directly on top of line x. Hint Line Segments Return to Table of Contents Line Segments or Line segments are portions of a line. or endpoint endpoint is read segment AB.. Line Segment segment. or are different names for the same It consists of the endpoints A and B 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. 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 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 A coordinate Distance AF = |-8 - 6| = 14 F 5 6 7 8 9 10 F coordinate X Why did we take the Absolute Value when calculating distance? 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 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 Find the measure of each segment in centimeters. cm a. CE = b. AB = 27 Find a segment that is 4 cm long cm A DA B BD C DE D CD 28 Find a segment that is 6.5 cm long cm A DA B BE C DE D CD 29 Find a segment that is 3.5 cm long cm A AC B BE C BD D DC 30 Find a segment that is 2 cm long cm A DE B CA C BD D DC 31 Find a segment that is 5.5 cm long cm A CE B EB C DB D DA 32 If point F was placed at 3.5 cm on the ruler, how from point E would it be? cm A 5 cm B 4 cm C 3.5 cm D 4.5 cm Segment Addition Postulate If B is between A and C, then AB + BC = AC. Or, said another way, If AB + BC = AC, then B is between A and C. AB BC AC 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. 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). In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 Find CD and BE Start by filling in the information you are given AE AB BC CD DE In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 27 || 6 Can you finish the rest? || 5 CD = BE = Example 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 1) Draw a diagram and insert the information given into the diagram 2) From the segment addition postulate, we know that KP + PM = MK (the parts equal the whole) 3) (x + 17) + (2x + 4) = 14x - 56 3x + 21 = 14x - 56 + 56 + 56 3x + 77 = 14x -3x - 3x 77 = 11x 7=x 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 For next group of questions (#31-36), we are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD Hint: always start these problems by placing the information you have into the diagram. 33 We are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD What is AB, BC, and CD? 34 We are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD What is DE? 35 We are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD What is CA? 36 We are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD What is CE? 37 We are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD What is DA? 38 We are given the following information about the collinear points: AE = 20 BD = 6 AB = BC = CD What is BE? 39 X, 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 40 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 41 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 The Pythagorean Theorem Return to Table of Contents 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 BC. The Pythagorean Theorem 2 2 c b 2 c =a +b 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. Proof Proof 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. 2 2 2 2 c =a +b 5 a=? 25 = a + 9 -9 -9 2 16 = a = a 4 = a 3 You will use the Pythagorean Theorem often. Example Answer 42 What is the length of side c? (The longest side of a triangle is called the ?) c2= a2 + b2 Answer 43 What is the length of side a? Hint: Always determine which side is the hypotenuse first 44 What is the length of c? B 45 What is the length of the missing side? 46 What is the length of side b? 47 What is the measure of x? x 8 17 Pythagorean Triples are three positive integers for side lengths that satisfy 2 2 2 a +b =c ( 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 48 A triangle has sides 30, 40 , and 50, is it a right triangle? Yes No 49 A triangle has sides 9, 15 , and 12, is it a right triangle? Yes No 50 A triangle has sides √3, 2 , and √5, is it a right triangle? Yes No Distance between Points Return to Table of Contents Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles. Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle. Relationship between the Pythagorean Theorem & Distance Formula The Pythagorean Theorem states a relationship among the sides of a right triangle. 2 2 The distance formula calculates the distance using point's coordinates. (x2, y2) 2 c =a +b c c a b (x1, y1) (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. Distance The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula. The Distance Formula The distance 'd' between any two points with coordinates (x1 ,y1) and (x2 ,y2) is given by the formula: d= Note: recall that all coordinates are (x-coordinate, y-coordinate). Example Calculate the distance from Point K to Point I (x2, y2) (x1, y1) 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 = = = 51 Calculate the distance from Point J to Point K A B C D 52 Calculate the distance from H to K A B C D 53 Calculate the distance from Point G to Point K A B C D 54 Calculate the distance from Point I to Point H A B C D 55 Calculate the distance from Point G to Point H A B C D Midpoint Formula Return to Table of Contents Midpoint of a line segment A number line can help you find the midpoint of a segment. The midpoint of GH, marked by point M, is -1. 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 Midpoint Formula Theorem The midpoint of a segment joining points with coordinates (x1, y1) and (x2, y2) is the point with coordinates 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. Just as before, we find the average of the coordinates. Remember that points are written with the x-coordinate first. (x, y) ( , ) The coordinates of M, the midpoint of PQ, are (6, 5) 56 Find the midpoint coordinates (x,y) of the segment connecting points A(1,2) and B(5,6) A (4, 3) B (3, 4) C (6, 8) D (2.5, 3) Hint: Always label the points coordinates first 57 Find the midpoint coordinates (x,y) of the segment connecting the points A(-2,5) and B(4, -3) A (-1, -1) B (-3, -8) C (-8, -3) D (1, 1) 58 Find the coordinates of the midpoint (x, y) of the segment with endpoints R(-4, 6) and Q(2, -8) A (-1, 1) B (1, 1) C (-1, -1) D (1, -1) 59 Find the coordinates (x, y) of the midpoint of the segment with endpoints B(-1, 3) and C(-7, 9) A (-3, 3) B (6, -4) C (-4, 6) D (4, 6) 60 Find the midpoint (x, y) of the line segment between A(-1, 3) and B(2,2) A (3/2, 5/2) B (1/2, 5/2) C (1/2, 3) D (3, 1/2) Example: Finding the coordinates of an endpoint of an segment Use the midpoint formula to write equations using x and y. 61 Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0) A (-1, -2) B (-2, -1) C (4, 2) D (2, 4) 62 Find the other endpoint of the segment with the endpoint (1, 4) and (5, -2) A (1, 3) B (3, 1) C (3, -3) D (-3, 3) Locus & Constructions Return to Table of Contents Introduction to Locus Definition: Locus is a set of points that all satisfy a certain condition, in this course, it is the set of points that are the same distance from something else. Theorem: locus between two points All points on the perpendicular bisector of a line segment connecting two points are equidistant from the two points. Theorem: locus from a given line All points equidistant from a given line is a parallel line. Theorem: locus between two lines The locus of points equidistant from two given parallel lines is a parallel line midway between them. locus: equidistant from two points The locus of points equidistant from two points, A and B, is the perpendicular bisector of the line segment determined by the two points. X Y The distance (d) from point A to the locus is equal to the distance (d') from Point B to the locus. The set of all these points forms the red line and is named the locus. Point M is the midpoint of . X is equidistant (d=d') from A and B. Y lies on the locus; it is also equidistant from A and B. Constructions Dividing a line segment into x congruent segments. Let us divide AB into 3 equal segments - we could choose any number of segments. 1. From point A, draw a line segment at an angle to the given line, and about the same length. The exact length is not important. 2. Set the compass on A, and set its width to a bit less than 1/3 of the length of the new line. Step the compass along the line, marking off 3 arcs. Label the last C. 3. Set the compass width to CB. 4. Using the compass set to CB, draw an arc below A 5. With the compass width set to AC, draw an arc from B intersecting the arc you just drew in step 4. Label this D. 6. Draw a line connecting B with D 7. Set the compass width back to AC and step along DB making 4 new arcs across the line 8. Draw lines connecting the arc along AC and BD. These lines intersect AB and divide it into 3 congruent segments. 63 Point C is on the locus between point A and point B True False 64 Point C is on the locus between point A and point B True False 65 How many points are equidistant from the endpoints of ? A 2 B 1 C 0 D infinite 66 You can find the midpoint of a line segment by A measuring with a ruler B constructing the midpoint C finding the intersection of the locus and line segment D all of the above 67 The definition of locus A a straight line between two points B the midpoint of a segment C the set of all points equidistant from two other points D a set of points Example: Construction Divide the line segment into 3 congruent segments. Angles & Angle Relationships Return to Table of Contents Identifying Angles An angle is formed by two rays with a common endpoint (vertex) The angle shown can be called When there is no chance of confusion, the angle may also be identified by its vertex B. The sides of are CB and AB , , or . C (Side) 32° B (Vertex) (Side) A 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. Angle Measures 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. A B is a 23° degree angle The measure of is 23° degrees C is a 119° degree angle The measure of is 119° degrees In and , notice that the vertex is written in between the sides L M Example K P N J O Challenge Questions X Angle Relationships Once we know the measurements of angles, we can categorize them into several groups of angles: 0° < acute < 90° right = 90° 90° < obtuse < 180° straight = 180° 180° 180° < reflex angle < 360° Two lines or line segments that meet at a right angle are said to be perpendicular. Link 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°) + = 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 non-shared sides form a line. A linear pair of angles are two adjacent angles whose noncommon sides on the same line. A line could also be called a straight angle with 180° Example Solution: Choose a variable for the angle - I'll choose "x" Example Let x = the angle Since the angles are complementary we know their sum must equal 90 degrees. 90 = 2x + x 90 = 3x 30 = x 68 An angle is 34° more than its complement. What is its measure? Hint: Choose a variable for the angle What is a complement? angle = complement + 34 angle = (90 - x) + 34 Answer x = 90 - x +34 2x = 124 x = 62 69 An angle is 14° less than its complement. What is the angle's measure? Hint: What is a complement? Choose a variable for the angle angle = complement - 14 angle = (90 - x) - 14 x = 90 - x - 14 Answer 2x = 90 - 14 2x = 76 x = 38 70 An angle is 98 o more than its supplement. What is the measure of the angle? Hint: Choose a variable for the angle What is a complement? angle = (180 - x) + 98 x Answer = 180 - x + 98 2x = 278 x = 139 71 An angle is 74° less than its supplement. What is the angle? angle = supplement - 74 x = (180 - x) - 74 Answer 2x = 180 - 74 2x = 106 x = 53 72 An angle is 26° more than its supplement. What is the angle? angle = supplement + 26 x = (180 - x) + 26 Answer 2x = 180 + 26 2x = 206 x = 106 Angle Addition Postulate Return to Table of Contents Angle Addition Postulate if a point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR. 32° 26° m∠PQS = 32° + m∠SQR = 26° 58° m∠PQR = 58° Just as from the Segment Addition Postulate, "The whole is the sum of the parts" Example 73 Given m∠ABC = 23° and m∠DBC = 46°. Find m∠ABD Hint: Always label your diagram with the information given 74 Given m∠OLM = 64° and m∠OLN = 53°. Find m∠NLM A 28 B 15 C 11 D 117 75 Given m∠ABD = 95° and m∠CBA = 48°. Find m∠DBC 76 Given m∠KLJ = 145° and m∠KLH = 61°. Find m∠HLJ 77 Given m∠TRQ = 61° and m∠SRQ = 153°. Find m∠SRT 78 Hint: Draw a diagram and label it with the given information 79 Hint: Draw a diagram and label it with the given information 80 Hint: Draw a diagram and label it with the given information Angle Bisectors & Constructions Return to Table of Contents Definition: Angle Bisector bisects Angle Bisector: A ray or line which starts at the vertex and cuts an angle into two equal halves Bisect means to cut it into two equal parts. The 'bisector' is the thing doing the cutting. The angle bisector is equidistant from the sides of the angle when measured along a segment perpendicular to the sides of the angle. Try this! Bisect the angle Try this! Bisect the angle