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Slide 1 / 168 Slide 2 / 168 New Jersey Center for Teaching and Learning 8th Grade Math 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. 2D Geometry: Transformations 2013-12-09 www.njctl.org Click to go to website: www.njctl.org Slide 3 / 168 Slide 4 / 168 Table of Contents Click on a topic to go to that section · Transformations · Translations Transformations · Rotations · Reflections · · · · Dilations Symmetry Congruence & Similarity Special Pairs of Angles Return to Table of Contents Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5 Slide 5 / 168 Slide 6 / 168 Any time you move, shrink, or enlarge a figure you make a transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on the transformed image (image) with the same letters and the prime sign. Pull image pre-image C C' B B' A' A Triangle ABC is the pre-image to the reflected image triangle XYZ for transformation shown B The image can also be labeled with new letters as shown below. Y X A image pre-image C Z Slide 7 / 168 Slide 8 / 168 There are four types of transformations in this unit: There are four types of transformations in this unit: · · · · Translations Rotations Reflections Dilations · · · · The first three transformations preserve the size and shape of the figure. They will be congruent. Congruent figures are same size and same shape. In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid. If your pre-image is an angle, your image is an angle with the same measure. If your pre-image contains parallel lines, your image contains parallel lines. Slide 9 / 168 Translations Rotations Reflections Dilations The first three transformations preserve the size and shape of the figure. In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid. If your pre-image is an angle, your image is an angle with the same measure. If your pre-image contains parallel lines, your image contains parallel lines. Slide 10 / 168 Translations Return to Table of Contents Slide 11 / 168 A translation is a slide that moves a figure to a different position (left, right, up or down) without changing its size or shape and without flipping or turning it. You can use a slide arrow to show the direction and distance of the movement. Slide 12 / 168 This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved right 7 and up 4. Slide 13 / 168 Slide 14 / 168 To complete a translation, move each point of the pre-image and label the new point. Click for web page B' A' PULL Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image? C' D' A D B C Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent. Slide 15 / 168 Slide 16 / 168 Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image? Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image? A PULL A PULL C B D B C Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent. Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent. Slide 17 / 168 Slide 18 / 168 A rule can be written to describe translations on the coordinate plane. Look at the following rules and coordinates to see if you can find a pattern. Translate pre-image ABCD 5 left and 3 up. What are the coordinates of the image and pre-image? PULL A B C D Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0) Slide 19 / 168 Slide 20 / 168 Translating left/right changes the x-coordinate. Translating left/right changes the x-coordinate. · Left subtracts from the x-coordinate Translating up/down changes the y-coordinate. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0) · Right adds to the x-coordinate Translating up/down changes the y-coordinate. · Down subtracts from the y-coordinate · Up adds to the y-coordinate Slide 21 / 168 Slide 22 / 168 Write a rule for each translation. A rule can be written to describe translations on the coordinate plane. 2 units Left … x-coordinate - 2 y-coordinate stays click to reveal rule = (x - 2, y) 5 units Right & 3 units Down… x-coordinate + 5 y-coordinate -3 click to reveal rule = (x + 5, y - 3) 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) click reveal (x, y) to(x-2, y+5) click reveal (x, y) to(x-2, y-6) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0) (x,click y) to (x-5, y+3) reveal reveal (x,click y) to (x+4, y-1) Slide 23 / 168 1 Slide 24 / 168 What rule describes the translation shown? A (x,y) E' (x - 4, y - 6) B (x,y) (x - 6, y - 4) C (x,y) (x + 6, y + 4) D (x,y) (x + 4, y + 6) 2 D' D F' E F G' Pull What rule describes the translation shown? A (x,y) (x, y - 9) B (x,y) (x, y - 3) C (x,y) (x - 9, y) D (x,y) (x - 3, y) D G D' E F G E' F' G' Pull Slide 25 / 168 3 Slide 26 / 168 What rule describes the translation shown? A (x,y) (x + 8, y - 5) B (x,y) (x - 5, y - 1) C (x,y) (x + 5, y - 8) D (x,y) (x - 8, y + 5) D' 4 Pull E' F' G' E D What rule describes the translation shown? A (x,y) (x - 3, y + 2) B (x,y) (x + 3, y - 2) C (x,y) (x + 2, y - 3) D (x,y) (x - 2, y + 3) D Pull E D' F E' F' G F G' G Slide 27 / 168 5 Slide 28 / 168 What rule describes the translation shown? A (x,y) (x - 3, y + 2) B (x,y) (x + 3, y - 2) C (x,y) (x + 2, y - 3) D (x,y) (x - 2, y + 3) Pull E' D' F' E D F Rotations G' G Return to Table of Contents Slide 29 / 168 Slide 30 / 168 A rotation (turn) moves a figure around a point. This point can be on the figure or it can be some other point. This point is called the point of rotation. P Slide 31 / 168 Slide 32 / 168 When you rotate a figure, you can describe the rotation by giving the direction (clockwise or counterclockwise) and the angle that the figure is rotated around the point of rotation. Rotations are counterclockwise unless you are told otherwise. Describe each of the rotations. Rotation Hint A The person's finger is the point of rotation for each figure. B This figure is rotated Click for answer 180 degrees clockwise about point B. This figure is rotated Click for answer 90 degrees counterclockwise about point A. Slide 33 / 168 How is this figure rotated about the origin? In a coordinate plane, each quadrant represents B A B' D A' C C' D' Slide 34 / 168 In order to determine the angle, draw two rays (one from the point of rotation to pre-image point, the other from the point of rotation to the image point). Measure this angle. The following descriptions describe the same rotation. What do you notice? Can you give your own example? This figure is rotated 270 degrees clockwise Click about the origin orto 90Reveal degrees counterclockwise about the origin. Check to see if the pre-image and image are congruent. Slide 35 / 168 The sum of the two rotations (clockwise and counterclockwise) is 360 degrees. If you have one rotation, you can calculate the other by subtracting from 360. Slide 36 / 168 6 How is this figure rotated about point A? (Choose more than one answer.) C A clockwise B counterclockwise C 90 degrees D 180 degrees E 270 degrees D C' B' E D' B A, A' E' Check to see if the pre-image and image are congruent. Pull Slide 37 / 168 7 Slide 38 / 168 How is this figure rotated about point the origin? (Choose more than one answer.) A clockwise B counterclockwise C 90 degrees D 180 degrees E 270 degrees A B D C Pull C' A' Pull B' A' C C' D' When rotated counter-clockwise, the x-coordinate is the opposite of the pre-image y-coordinate and the y-coordinate is the same as the pre-image of the x-coordinate. In other words: Click to Reveal (x, y) Slide 39 / 168 Write the coordinates for the pre-image and image. D What do you notice? Check to see if the pre-image and image are congruent. What happens to the coordinates in a half-turn? B A Write the coordinates for the pre-image and image. D' B' Now let's look at the same figure and see what happens to the coordinates when we rotate a figure. (-y, x) Slide 40 / 168 Can you summarize what happens to the coordinates during a rotation? A D What do you notice? B Pull Counterclockwise: (x, y)to Reveal (-y, x) Click C C' B' D' Half-turn: (x, y) to Reveal (-x, -y) Click A' Clockwise: (x, y) (y, -x) Click to Reveal When rotated a half-turn, the x-coordinate is the opposite of the pre-image x-coordinate and the y-coordinate is the opposite of the pre-image of the y-coordinate. other words: Click toInReveal (x, y) (-x, -y) Slide 41 / 168 Slide 42 / 168 9 What are the new coordinates of a point A (5, -6) after a rotation clockwise? 8 A (-6, -5) B (6, -5) Pull What are the new coordinates of a point S (-8, -1) after a rotation counterclockwise? A (-1, -8) B (1, -8) C (-5, 6) C (-1, 8) D (5, -6) D (8, 1) Pull Slide 43 / 168 Slide 44 / 168 What are the new coordinates of a point H (-5, 4) after a rotation counterclockwise? 10 A (-5, -4) 11 Pull B (5, -4) What are the new coordinates of a point R (-4, -2) after a rotation clockwise? A (2, -4) Pull B (-2, 4) C (4, -5) C (2, 4) D (-4, 5) D (-4, 2) Slide 45 / 168 Slide 46 / 168 12 What are the new coordinates of a point Y (9, -12) after a half-turn? A (-12, 9) Pull B (-9,12) C (-12, -9) D (9,12) Reflections Return to Table of Contents Slide 47 / 168 Slide 48 / 168 Examples A reflection (flip) creates a mirror image of a figure. Slide 49 / 168 Slide 50 / 168 A reflection is a flip because the figure is flipped over a line. Each point in the image is the same distance from the line as the original point. t A and A' are both 6 units from line t. B and B' are both 6 units from line t. C and C' are both 3 units from line t. A A' B C y A B D C Each vertex in ABC is the same distance from line t as the vertices in A'B'C'. B' C' Reflect the figure across the y-axis. Check to see if the pre-image and image are congruent. x Check to see if the pre-image and image are congruent. Slide 51 / 168 Slide 52 / 168 What do you notice about the coordinates when you reflect across the y-axis? y A D B C Tap box for coordinates B' A (-6, 5) B (-4, 5) C (-4, 1) D (-6, 3) A' D' C' x A' (6, 5) B' (4, 5) C' (4, 1) D' (6, 3) When you reflect across the y-axis, the x-coordinate becomes the opposite. So (x, y) (-x, y) when you reflect across the yaxis. Check to see if the pre-image and image are congruent. What do you predict about the coordinates when you reflect across the x-axis? Tap box for coordinates y A D A (-6, 5) B (-4, 5) C (-4, 1) D (-6, 3) B C D' A' B' So (x, y) (x, -y) when you reflect across the xaxis. Check to see if the pre-image and image are congruent. Slide 53 / 168 Reflect the figure across the y-axis then the x-axis. Click to see each reflection. y Slide 54 / 168 Reflect the figure across the y-axis. Click to see reflection. y A A B B C D When you reflect across the x-axis, the y-coordinate becomes the opposite. x C' A' (-6, -5) B' (-4, -5) C' (-4, -1) D' (-6, -3) C x Check to see if the pre-image and image are congruent. F D E x Check to see if the pre-image and image are congruent. Slide 55 / 168 Slide 56 / 168 Reflect the figure across the line x = -2. Reflect the figure across the line y = x. y y B C A E x D Check to see if the pre-image and image are congruent. A B D C Check to see if the pre-image and image are congruent. Slide 57 / 168 Slide 58 / 168 13 The reflection below represents a reflection across: 14 The reflection below represents a reflection across: A the x axis C the x axis, then the y axis A the x axis B the y axis D the y axis, then the x axis B the y axis y y B B' A Pull A the x axis, then the y axis D the y axis, then the y axis D C B C x C' C' Slide 60 / 168 Which of the following represents a single reflection of Figure 1? 15 16 Which of the following describes the movement below? A Figure 1 D A' Check to see if the pre-image and image are congruent. Slide 59 / 168 C x B' D' Check to see if the pre-image and image are congruent. B C Pull A' A x Pull reflection B rotation, 90 clockwise C slide D rotation, 180 clockwise Pull Slide 61 / 168 Slide 62 / 168 17 Describe the reflection below: 18 Describe the reflection below: A across the line y = x C across the line y = -3 A across the line y = x C across the line y = -3 B across the y axis D across the x axis B across the x axis D across the line x = 4 y Pull D' C' E' B' A' Pull y A' A B C B x A E B' C C' x D Check to see if the pre-image and image are congruent. Slide 63 / 168 Check to see if the pre-image and image are congruent. Slide 64 / 168 Dilations Return to Table of Contents Slide 65 / 168 Slide 66 / 168 The scale factor is the ratio of sides: A dilationis a transformation in which a figure is enlarged or reduced around a center point using a scale factor = 0. The center point is not altered. When the scale factor of a dilation is greater than 1, the dilation is an enlargement . When the scale factor of a dilation is less than 1, the dilation is a reduction. When the scale factor is |1|, the dilation is an identity. Slide 67 / 168 Slide 68 / 168 Example. What happened to the coordinates with a scale factor of 2? If the pre-image is dotted and the image is solid, what type of dilation is this? What is the scale factor of the dilation? y y x A (0, 1) B (3, 2) C (4, 0) D (1, 0) B' This is an enlargement. Scale Factor is 2. A' A to reveal base length Click of image base length of pre-image B D D' C 6 3 = 2 C' x A' (0, 2) B' (6, 4) C' (8, 0) D' (2, 0) The coordinates were all Click to reveal multiplied by 2. The center for this dilation was the origin (0,0). Slide 69 / 168 Slide 70 / 168 20What are the coordinates of a point S (3, -2) after a dilation with a scale factor of 4 about the origin? 19What is the scale factor for the image shown below? The pre-image is dotted and y the image is solid. A 2 A (12, -8) B 3 Pull C -3 B (-12, -8) C (-12, 8) D 4 D (-3/4, 1/2) x Slide 71 / 168 Slide 72 / 168 22What are the coordinates of a point X (4, -8) after a dilation with a scale factor of 0.5? 21What are the coordinates of a point Y (-2, 5) after a dilation with a scale factor of 2.5? A (-8, 16) A (-0.8, 2) B (-5, 12.5) Pull Pull B (8, -16) C (0.8, -2) C (-2, 4) D (5, -12.5) D (2, -4) Pull Slide 73 / 168 The coordinates of a point change as follows during a dilation: (-6, 3) (-2, 1) 23 Slide 74 / 168 24The coordinates of a point change as follows during a dilation: (4, -9) (16, -36) Pull What is the scale factor? What is the scale factor? A 3 A 4 B -3 B -4 C 1/3 C 1/4 D -1/3 D -1/4 Pull Slide 75 / 168 Slide 76 / 168 26 25The coordinates of a point change as follows during a dilation: (5, -2) (17.5, -7) Which of the following figures represents a rotation? (and could not have been achieved only using a reflection) A Figure A B Figure B Pull What is the scale factor? Pull A 3 B -3.75 C Figure C C -3.5 D Figure D D 3.5 Slide 77 / 168 27Which Slide 78 / 168 of the following figures represents a reflection? A Figure A B 28 Figure B Which of the following figures represents a dilation? B Figure B A Figure A Pull Pull C Figure C D Figure D C Figure C D Figure D Slide 79 / 168 29 Slide 80 / 168 Which of the following figures represents a translation? B Figure B A Figure A Pull C Figure C D Symmetry Figure D Return to Table of Contents Slide 81 / 168 Slide 82 / 168 Symmetry A line of symmetry divides a figure into two parts that match each other exactly when you fold along the dotted line. Draw the lines of symmetry for each figure below if they exist. Slide 83 / 168 Which of these figures have symmetry? Draw the lines of symmetry. Slide 84 / 168 Do these images have symmetry? Where? Slide 85 / 168 Slide 86 / 168 We think that our faces are symmetrical, but most faces are asymmetrical (not symmetrical). Here are a few pictures of people if their faces were symmetrical. Will Smith with a symmetrical face. Click the picture below to learn how to make your own face symmetrical. Marilyn Monroe with a symmetrical face. Tina Fey Slide 87 / 168 Slide 88 / 168 Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn. Rotate these figures. What degree of rotational symmetry do each of these figures have? Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn. Slide 89 / 168 30 Slide 90 / 168 How many lines of symmetry does this figure have? 31Which A 3 B 6 C 5 D 4 Pull A figure's dotted line shows a line of symmetry? B C D Pull Slide 91 / 168 32 Slide 92 / 168 Which of the objects does not have rotational symmetry? A Pull B C Congruence & Similarity D Return to Table of Contents Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn. Click for hint. Slide 93 / 168 Congruence and Similarity Slide 94 / 168 Similar shapes have the same shape, congruent angles and proportional sides. Congruent shapes have the same size and shape. 2 figures are congruent if the second figure can be obtained from the first by a series of translations, reflections, and/or rotations. Pull 2 figures are similar if the second figure can be obtained from the first by a series of translations, reflections, rotations and/or dilations. Remember - translations, reflections and rotations preserve image size and shape. Slide 95 / 168 Slide 96 / 168 Example Click for web page What would the measure of angle j have to be in order for the figures below to be similar? j 180 - 112 - 33 = 35 Slide 97 / 168 Slide 98 / 168 Example 33Which Are the two triangles below similar? Explain your reasoning? pair of shapes is similar but not congruent? A Pull B C Yes, the triangles have congruent angles and areClick therefore similar. D Slide 99 / 168 34Which Slide 100 / 168 Which of the following terms best describes the pair of figures? pair of shapes is similar but not congruent? 35 A B Pull A congruent B similar C neither congruent nor similar Pull C D Slide 101 / 168 Slide 102 / 168 Which of the following terms best describes the pair of figures? 37 36Which of the following terms best describes the pair of figures? A congruent A congruent B similar B similar C neither congruent nor similar C neither congruent nor similar Pull Pull Slide 103 / 168 Slide 104 / 168 Determine if the two figures are congruent, similar or neither. Determine if the two figures are congruent, similar or neither. Be able to explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid. Be able to explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid. Pull Pull Slide 105 / 168 Slide 106 / 168 Determine if the two figures are congruent, similar or neither. Determine if the two figures are congruent, similar or neither. Be able to explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid. Be able to explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid. Pull Pull Slide 107 / 168 Slide 108 / 168 Recall: · Complementary Angles are two angles with a sum of 90 degrees. These two angles are complementary angles because their sum is 90. Special Pairs of Angles Notice that they form a right angle when placed together. · Supplementary Angles are two angles with a sum of 180 degrees. Return to Table of Contents These two angles are supplementary angles because their sum is 180. Notice that they form a straight angle when placed together. Slide 109 / 168 Slide 110 / 168 Transformations Vertical Angles are two angles that are opposite each other when two lines intersect. a b d Vertical Angles can further be explained using the transformation of reflection. b c c a d x In this example, the vertical angles are: Line x cuts angles b and d in half. Vertical angles have the same measurement. So: When angle a is reflected over line x, it forms angle c. When angle c is reflected over line x, it forms angle a. Slide 111 / 168 Slide 112 / 168 Transformations Continued Using what you know about complementary, supplementary and vertical angles, find the measure of the missing angles. y b c a d b c a Line y cuts angles a and c in half. By Vertical Angles: When angle b is reflected over line y, it forms angle d. Click When angle d is reflected over line y, it forms angle b. Slide 113 / 168 38Are By Supplementary Angles: Click Slide 114 / 168 angles 2 and 4 vertical angles? 39 Yes Are angles 2 and 3 vertical angles? Yes No No 1 2 4 3 Pull 3 2 4 1 Pull Slide 115 / 168 40 Slide 116 / 168 41 If angle 1 is 60 degrees, what is the measure of angle 3? You must be able to explain why. A 30 o B 60 C 120 A 30 o Pull o o B 60 o C 120 2 D 15 o 1 4 If angle 1 is 60 degrees, what is the measure of angle 2? You must be able to explain why. Slide 117 / 168 3 Adjacent or Not Adjacent? You Decide! A a is adjacent to b How do you know? · They have a common side (ray ) · They have a common vertex (point B) a a click to reveal Adjacent b b clickAdjacent to reveal Not click to reveal Not Adjacent D B Slide 119 / 168 42 4 Slide 118 / 168 Adjacent Angles are two angles that are next to each other and have a common ray between them. This means that they are on the same plane and they share no internal points. C 1 D 15 o 3 Pull 2 o Slide 120 / 168 43 Which two angles are adjacent to each other? A 1 and 4 B 2 and 4 Which two angles are adjacent to each other? A 3 and 6 B 5 and 4 Pull Pull 1 4 2 5 6 3 2 4 1 3 6 5 Slide 121 / 168 Slide 122 / 168 Recall From 3rd Grade A transversal is a line that cuts across two or more (usually parallel) lines. Shapes and Perimeters Parallel lines are a set of two lines that do not intersect (touch). A P E Q F A R B Interactive Activity-Click Here Slide 123 / 168 Slide 124 / 168 44 T ra nsv e rs al Corresponding Angles are on the same side of the transversal and on the same side of the given lines. In this diagram the corresponding angles are: a c d b Which are pairs of corresponding angles? A 2 and 6 B 3 and 7 C 1 and 8 Pull 1 3 5 f e g 7 2 4 6 8 h Click Slide 125 / 168 45 Slide 126 / 168 46 Which are pairs of corresponding angles? A 2 and 6 B 3 and 1 C Which are pairs of corresponding angles? A 1 and 5 Pull 1 and 8 6 2 1 7 2 1 4 3 C 4 and 8 5 7 4 8 3 5 B 2 and 8 6 8 Pull Slide 127 / 168 Slide 128 / 168 47 Which are pairs of corresponding angles ? A 2 and 4 5 4 C 7 and 8 2 In this diagram the alternate interior angles are: a c d b m 6 3 7 D 1 and 3 Pull 1 8 B 6 and 5 Alternate Interior Angles are on opposite sides of the transversal and on the inside of the given lines. l g n f e h Click Slide 129 / 168 Slide 130 / 168 Alternate Exterior Angles are on opposite sides of the transversal and on the outside of the given lines. l Same Side Interior Angles are on same sides of the transversal and on the inside of the given lines. l In this diagram the same side interior angles are: a c g In this diagram the alternate exterior angles are: m a c n f e Click d b g Click Slide 131 / 168 48 49 l Pull No 3 1 5 2 6 8 4 7 h m n Which line is the transversal? Slide 132 / 168 Are angles 2 and 7 alternate exterior angles? Yes b f e h d Are angles 3 and 6 alternate exterior angles? l Yes Pull No 3 1 m 5 n 2 6 8 4 7 m n Slide 133 / 168 50 Slide 134 / 168 Are angles 7 and 4 alternate exterior angles? 51 Which angle corresponds to angle 5? l Yes No 3 5 2 6 C m 2 6 Slide 135 / 168 52Which 53 Pull B C 3 1 D 5 What type of angles are What type of angles are 7 4 l Pull B Alternate Exterior Angles Same Side Interior 2 3 m 7 4 n 8 What type of angles are and ? A Alternate Interior Angles 5 Same Side Interior 2 6 8 4 1 3 7 C Corresponding Angles m 1 D Vertical Angles E n Pull l B Alternate Exterior Angles D Vertical Angles E 1 5 6 55 C Corresponding Angles Pull l Slide 138 / 168 ? A Alternate Interior Angles ? D Vertical Angles n 8 and and C Corresponding Angles m Slide 137 / 168 54 n B Alternate Exterior Angles E 2 4 8 A Alternate Interior Angles l A m 7 Slide 136 / 168 pair of angles are same side interior? 6 5 n 8 3 1 D 7 4 Pull B Pull 1 l A 5 Same Side Interior 2 6 8 4 3 7 m n Slide 139 / 168 56 Slide 140 / 168 Are angles 5 and 2 alternate interior angles? 57 l Yes No Pull 1 3 5 7 2 l Yes No 6 Are angles 5 and 7 alternate interior angles? Pull 1 m 5 4 2 n 8 6 4 n Slide 142 / 168 Are angles 7 and 2 alternate interior angles? 59 l Yes m 8 Slide 141 / 168 58 3 7 Are angles 3 and 6 alternate exterior angles? l Yes No No 3 1 5 2 6 Pull m 7 5 n 4 2 6 8 Pull 3 1 m 7 4 n 8 Slide 143 / 168 Slide 144 / 168 Special Cases Reflections If parallel lines are cut by a transversal then: n · Corresponding Angles are congruent 1 5 · Alternate Interior Angles are congruent 2 · Same Side Interior Angles are supplementary SO: 6 n 1 click 5 are supplementary 6 3 m 7 4 2 7 b · Alternate Exterior Angles are congruent are supplementary a m 3 8 These Special Cases can further be explained using the transformations of reflections and translations l 4 l 8 Line a cuts angles 3 and 5 in half. Line b cuts angles 4 and 6 in half. When angle 1 is reflected over line a, it forms angle 7. When angle 2 is reflected over line b, it forms angle 8. When angle 7 is reflected over line a, it forms angle 1. When angle 8 is reflected over line b, it forms angle 2. Slide 145 / 168 Slide 146 / 168 Reflections Continued 1 1 m 3 5 Translations n n c 5 m 3 7 Line m is parallel to line l. 7 d 2 6 2 l 4 6 l 4 8 8 n Line c cuts angles 1 and 7 in half. Line d cuts angles 2 and 8 in half. When angle 3 is reflected over line c, it forms angle 5. When angle 4 is reflected over line d, it forms angle 6. 2 6 When angle 5 is reflected over line c, it forms angle 3. 1 4 8 5 Slide 148 / 168 Translations Continued If line m is then translated x units left, all angles formed by lines m and n will overlap with all angles formed by lines l and n. l m 43 Given the measure of one angle, find the measures of as many angles as possible. Which angles are congruent to the given angle? 60 n 1 2 ml 7 When angle 6 is reflected over line d, it forms angle 4. Slide 147 / 168 56 3 If line m is translated y units down, it will overlap with line l. l A <4, <5, <6 B <5, <7, <1 4 C <2 78 5 m 6 Pull D <5, <1 n 12 56 43 2 The translations also work if line l is translated y units up and x units right. l m 78 1 Slide 149 / 168 Given the measure of one angle, find the measures of as many angles as possible. What are the measures of angles 4, 6, 2 and 8? Given the measure of one angle, find the measures of as many angles as possible. Which angles are congruent to the given angle? 4 5 6 o A <4 l B <4, <5, <3 m Pull C 130 62 l B 40 o n Slide 150 / 168 61 A 50 o 7 8 5 D <8 2 1 2 7 8 n 3 1 C <2 8 4 Pull m 7 n Slide 151 / 168 Slide 152 / 168 64 Given the measure of one angle, find the measures of as many angles as possible. What are the measures of angles 2, 4 and 8 respectively? 63 B 35 o , 35 o , 35 o C 145 o , 35 o , 145 1 4 l o 3 1 5 2 2 a 3 4 B Translation Only n 8 7 8 A Reflection Only 7 b 6 5 m Pull 0 t Pull A 55 o , 35 o , 55 If lines a and b are parallel, which transformation justifies why ? C Reflection and Translation D The Angles are NOT Congruent Slide 153 / 168 If lines a and b are parallel, which transformation justifies why ? 66 If lines a and b are parallel, which transformation justifies why ? t t 4 2 5 A Reflection Only 1 a 3 2 5 A Reflection Only 7 D The Angles are NOT Congruent b 6 8 7 B Translation Only B Translation Only C Reflection and Translation a 3 b 6 8 4 Pull 1 Pull 65 Slide 154 / 168 C Reflection and Translation D The Angles are NOT Congruent Slide 155 / 168 Slide 156 / 168 We can use what we've learned to establish some interesting information about triangles. For example, the sum of the angles of a triangle = 180. Let's see why! Applying what we've learned to prove some interesting math facts... Given B A C Slide 157 / 168 Slide 158 / 168 Let's draw a line through B parallel to AC. We then have a two parallel lines cut by a transversal. Number the angles and use what you know to prove the sum of the measures of the angles equals 180. l 1 B 2 C A l 1 B m p n 1. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 2 A C m p n 2. is supplementary with since if 2 parallel lines are cut by a transversal, then same side interior angles are supplementary. 3. Therefore, Slide 159 / 168 Slide 160 / 168 Let's look at this another way... 2 B Let's prove the Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the remote interior angles. l 1 C A B m p n 1 1. and since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. Remote Interior Angles Slide 161 / 168 Slide 162 / 168 Let's draw a line through B parallel to AC. We then have a two parallel lines cut by a transversal. Number the angles and use what you know to prove the measure of angle 1 = the sum of the measures of angles B and C. B l 2 3 1 l 2 C Exterior Angle 2. Since all three angles form a straight line, the sum of the angles is B A n A C m p 1. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 1 n A C m p 2. 3. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 4. Therefore, Slide 163 / 168 Slide 164 / 168 Example Example What is the measure of angle v in the diagram below? What angles are congruent to angle 9? 1 6 2 3 p 5 4 v 10 7 9 11 12 14 13 8 r h g Click Slide 165 / 168 Slide 166 / 168 Example 67 What is the measure of angle q in the diagram below? Name the pairs of angles whose sum is congruent to angle 9. 10 7 9 3 p 5 4 q 11 12 14 13 8 r Pull 2 1 6 h g Click and Slide 167 / 168 Slide 168 / 168 the expression that will make the statement below true: What is the measure of angle 7? 2 10 g 7 9 8 2 3 p 5 4 11 12 14 13 h A C B D 6 Pull 1 6 10 r g 7 9 8 p 5 4 11 12 13 h r Pull 69 68 Choose