Butterflies, Pinwheels, and Wallpaper

Transcription

1 Butterflies, Pinwheels, and Wallpaper Investigation #3: Transforming Coordinates Investigation #4: Dilations and Similar Figures Name

2 Butterflies, Pinwheels and Wallpaper Investigation #3 Transforming Coordinates Investigation #4 Dilations and Similar Figures Problem 3.1: How can you describe how points move under a reflection with coordinate rules in the form (x, y) (, ) when the reflection line is: (1) the x-axis? (2) the y-axis? (3) the line y = x? Problem 3.2: What kind of coordinate rule (x, y) (, ) tells how to move any point to its image under a translation? Problem 3.3: What are the coordinate rules that describe motion of points on a grid under turns of 90 and 180? Problem 3.4: How are lines and their images under translations and half-turns related to each other? Problem 3.5: When two parallel lines are cut by a transversal, what can be said about the angles formed? What is always true about the angle measures in a triangle? How do you know that your answers are correct? Problem 4.1: What coordinate rules model dilations? How do dilations change or preserve characteristics of the original figure? Problem 4.2: How can you use transformations to check whether two figures are similar or not? 2 P a g e

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4 Reflecting over the y-axis: 4 P a g e

5 Are the two figures congruent? Explain. 5 P a g e

6 Reflecting over the x-axis: 6 P a g e

7 Are the two figures congruent? Explain. 7 P a g e

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9 Practice: 1. Quadrilateral ABCD is being reflected over the x-axis. Draw and label the image quadrilateral A B C D. We can use an arrow to describe this reflection. ABCD A B C D What are the coordinates of ABCD? A B C D A B C D 2. Describe a reflection that would move shape 1 to match shape 2. 9 P a g e

10 Key Concepts: A transformation is a change in the,, or of a figure. A reflection is a transformation which the figure over a. This line is called the. The original image of a transformation is called the. The resulting figure after a transformation is called the of the original figure, and is labeled with. Example: If a reflection occurs the image and pre-image are because the and are persevered. 10 P a g e

11 Homework: 1. a. Draw and label the image of the triangle HOT after a reflection over the x- axis. 7 y b. Write the coordinates of H O T O H x 1 T c. Is the image congruent to the preimage? How do you know? a. Draw and label the image of t quadrilateral WXYZ after a reflection over the y dotted line. Label the image W X Y Z b. Are the two figures congruent? X c. What is the equation of the dotted line? 1 W x Z 4 Y P a g e

12 3. The table below shows the coordinates of triangle PQR. Reflect PQR over the y- axis. Complete the table for the coordinates of the image. Triangle PQR P (-3, 2) P Q (3, 6) Q R (-7, 7) R Triangle P Q R Explain how the coordinates of R changed to the coordinates of R. 4. a) Draw ΔJKL which has coordinates J (-2,0), K (-3,4), and L (5,-1). b) Draw the image ΔJ K L after a reflection of ΔJKL over the x-axis. c) List the coordinates of J K L. J (-2,0) J K (-3, 4) K L (5, -1) L d) How did the coordinates of K change to the coordinates of K? e) Are the two figures congruent? Explain. 12 P a g e

13 Ex:) Given ΔABC and image ΔA B C, list the coordinates of image ΔA B C. A A B B C C Write the rule that translates ΔABC to its image Rule: (x, y) (, ) 13 P a g e

14 Part A: Make a table of the coordinates of key points for Mug 1 and his images under the translations. Look for patterns: 4. Are all the mugs congruent? Explain. E. Suppose a translation moves a figure a units horizontally and b units vertically on a coordinate grid. What rule describes the coordinates of each image point? 14 P a g e

15 Key Concepts: A translation is a transformation which each point of a figure the same and in the same. If a translation occurs the image and pre-image are because the and are persevered. Practice: 1. Write a general rule which describes the translation shown below. ΔLMN is the original triangle. In words: Arrow Notation: (x, y) (, ) 2. a) Graph points T(0,3), U(2, 4) and V(5, -1) and connect the points to make a triangle. b) Translate ΔTUV using the rule (x, y) (x - 3, y 1). c) In words, describe what the rule is asking you to do. d) Draw the image ΔT U V. e) Identify the coordinates of ΔT U V. T U V f) Using the image of ΔT U V perform an additional translation using the rule: (x, y) (x + 3, y - 3). Draw and Label ΔT U V g) Is this new image (ΔT U V ) congruent or similar to the original figure (ΔTUV)? Explain. 15 P a g e

16 Practice In questions a and b below, use arrow notation to write a rule (in words or arrow notation) that describes the translation shown on the graph. 1) 2) 3) Multiple Choice: Write a description of the rule. (x, y) (x 7, y + 4) 1. translation 7 units to the right and 4 units up. 2. translation 7 units to the left and 4 units down. 3. translation 7 units to the right and 4 units down. 4. translation 7 units to the left and 4 units up. 4) Refer to the grid below. a. Describe how you could move shape 1 to exactly match shape 2 by using one translation and one reflection. b. Are there other sequences of transformations that would move shape 1 to exactly match shape 2? If so, describe the steps you would perform. 16 P a g e

17 Homework 1. On the graph below, draw and label the image of triangle HOT after a translation six units left and one unit down. Is the image similar or congruent? How do you know? 7 y O H x T 2. On the graph below, draw and label the image of quadrilateral WXYZ under ( x, y) x 2, y 4. the rule 8 y X 1 W x Z Y P a g e

18 Use the grid below to answer questions 3 through Find the rule to describe the translation from point A to point B. 4. Find the rule to describe the translation from point C to point D. y 8 E A 6 4 C x 2 5. Find the rule to describe the translation from point E to point A. B D 6. Quadrilateral ABCD is plotted on the grid below. Part A: On the graph, draw the translation of polygon ABCD eight units to the left and seven units down. Label the image A B C D. A A B B C C D D Part B: Explain how you determined the location of A. 18 P a g e

19 7. Quadrilateral PQRS is plotted on the grid. y On the graph, draw the translation of polygon PQRS three units to the left and four units down. Label the image P Q R S. S P Q Now create polygon P Q R S by translating polygon P Q R S using the rule x, y x 2, y 1. What will be the coordinates of point Q? O R x Answer Write a single translation rule from polygon PQRS to polygon P Q R S. Triangle ABC is translated to create triangle DFG, as shown below: In these triangles, side AB is congruent to side DF, and side BC is congruent to side FG. Determine the values of x and y. 19 P a g e

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22 Are the two figures congruent? Explain. 22 P a g e

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24 3. Rotate Triangle ABC, 270 o counterclockwise. Label the triangle A B C. 4. Graph ABCD with vertices A(0, 4), B(2, 2), C(4, 2), and D(4, 4). What are the coordinates of the image of quadrilateral ABCD under a 90 clockwise rotation about the origin? A (0, 4) A B (2, 2) B C (4, 2) C D (4, 4) D 5. Determine the transformation that produced the following images: y a) b) y F' F' D' H' H' O H x O H x F D' F D D 24 P a g e

25 Key Concept: A turning of a geometric figure about a point is a. The is the point about which the figure is turned. The number of degrees the figure turns is the of rotation. If a rotation occurs the image and pre-image are because the and are persevered. Summary: A is when a reflection, translation or rotation occurs. If two figures are Congruent then the and are preserved. If a,, occur, then the two figures are congruent. 25 P a g e

26 1. a) Graph Triangle RST with vertices R 2, 3, S 5, 4, and T 4, 8. b) Rotate Triangle R S T 90 counterclockwise on the graph to the right. Label the new coordinates of the vertices of Triangle R S T. Graph the image. R (-2,3) R S (5,4) S T (4,8) T c) Are RST and R S T congruent to each other, similar to each other, or neither? Explain. 2. a.) Graph Triangle ABC, A(3, 2), B( 5, 4), C( 3, 2). b) Graph and label the image of Triangle ABC after a 180 counterclockwise rotation about the origin. A (3, 2) A B ( 5, 4) B C (3, 2) C 26 P a g e

27 3. Quadrilateral ABCD is plotted on the grid below. Part A On the graph, draw the image of quadrilateral ABCD after a counterclockwise rotation of 180 o about the origin. Label the image A B C D. List the coordinates below. A A B B C C D D Part B Explain how the coordinates of A changed to the coordinates of A. 27 P a g e

28 4. Point A (3, 6) is rotated 270 counterclockwise about the origin, what is the coordinate of A? Circle the best answer. (a) (b) 6, 3 (c) 6, 3 3, 6 (d) 3, 6 5. Point J (-2, 5) is rotated 90 counterclockwise about the origin. What is the coordinate of J? (a) (b) 2, 5 (c) 5, 2 5, 2 (d) 5, 2 6. Point D (1, 3) is rotated 180 about the origin. What is the coordinate of D? (a) (b), 1, 3 (c) 1, (d) 1, 3 7. What single transformation is equivalent to a reflection in the x-axis, followed by a reflection in the y-axis? 28 P a g e

29 8. Draw the final image created by rotating polygon ABCD 90 counterclockwise about the origin and then reflecting the image in the x-axis. Is the resulting image similar or congruent? How do we know? 9. Figure 1 can be transformed to create Figure 2 using a single transformation. Which transformation can be used to accomplish this? A Dilation C Reflection B Rotation D Translation 29 P a g e

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36 After a dilation occurs are the two figures congruent? Key Concepts: Dilation - transformation that produces an image that is the as the original but. A dilation is to the original figure. Dilations are centered around the origin (0, 0), unless otherwise stated. Scale factor: image length pre- image length, which is a. If the scale factor is greater than 1, the figure is an. If the scale factor is between 0 and 1, the figure is a. Similar Figures- Two figures are said to be similar if their are congruent and their are proportional. 36 P a g e

37 Practice: 1. Triangle ABC has vertices A (0, 2), B (4, 4), and C (-1, 4). What are the vertices of its image with a scale factor of 2? A ( ) B C y What is the rule for this dilation? x 1 2 ( x, y), 3 4 Graph the triangle and its image Quadrilateral ABCD has vertices A (-6, 3), B (3, 3), C (3, -3), and D(- 6, - 6). It is dilated by a scale factor of 1/3 What are the coordinates of the image? Graph the quadrilateral and its image. 37 P a g e

38 3. Which of the following describes the image of a figure after a dilation that has a scale factor between zero and one? a) It has a different shape from the original figure and is smaller than the original figure. b) It has the same shape as the original and is larger than the original figure. c) It has the same shape as the original and is smaller than the original figure. d) It has the same shape and same size as the original figure. 4. Which of the following describes the image of a square after a dilation that has a scale factor of 6? a) Its sides are 6 units longer than those of the original square. b) Its sides are 6 1 as long as those of the original square. c) Its sides are 6 times as long as those of the original square. d) Its sides are 6 units shorter than those of the original square. 5. Which of the following describes the image of a triangle after a dilation that has 5 a scale factor of? 6 5 a) Each angle has of the measure of its corresponding angle in the 6 original triangle. b) Each angle has 5 6 of the measure of its corresponding angle in the original triangle. c) Each angle has the same measure as its corresponding angle in the original triangle. d) Each angle is 6 1 larger than the measure of its corresponding angle in the original triangle. 38 P a g e

39 2 6. What are the coordinates of PQR after a dilation with a scale factor of? 3 a) P 2,1, Q 0,2, R 2,2 b) P 4,2, Q 0,4, R 4,4 c) P 4,2, Q 4,0, R 4,2 d) P 12,6, Q 0,12, R 12,12 7. D E F is the image of DEF after a dilation with a scale factor of 2. What are the coordinates of the vertices of DEF? a) D 8, 12, E 8,4, F 4, 4 b) D 6,4, E 2,0, F 4, 4 c) D 2,8, E 6,4, F 0,0 d) D 2,3, E 2,1, F 1, 1 39 P a g e

40 8. In the diagram below, ABC is similar to ART. Part A: What is the scale factor from ABC to ART? Part B: If the slope of AC is -2, what is the value of x for coordinate C? Part C: Using the information from parts A and B, what is the length of RT? 9. In the coordinate plane below, ABC is similar to AEF. What is the value of x? 40 P a g e

41 Practice: 1. If a scale factor is 2 5, how would you write the general rule? Is this an enlargement or a reduction? 2. Quadrilateral A B C D is a dilation of quadrilateral ABCD. Find the scale factor. Classify the dilation as an enlargement or a reduction. 3. Triangle XYZ is graphed below. a.) Draw and label Triangle X Y Z after a dilation using a scale factor of two y X (, ) X ', Y Z Y x 1 X Z b.) Are the two triangles similar? Explain how you know P a g e

42 4. Quadrilateral PQRS has vertices P (-2, 4), Q (4, 4), R (4, -2), and S (- 4, - 4). It is dilated by a scale factor of ½. What are the coordinates of the image? Graph the quadrilateral and its image. Are the two quadrilaterals similar? Explain. 5. Triangle XYZ has vertices X (0, -2), Y (-1, 2), and Z (2, 2). What are the vertices of its image with a scale factor of 3? y X ( ) Y Z What is the rule for this dilation? x Graph the triangle and its image. Are the two triangles similar? Explain P a g e

43 6. Triangle PQR has coordinates P 2,4, Q 2,4, R 0, 6. Write the coordinates of the vertices of the image of a triangle after a dilation of How does the size of an image compare to the original figure when the original figure undergoes a dilation with a scale factor of one? 8. On the grid below, draw the image of FGH after a dilation with a scale factor of 2 1. What will be the coordinates of point F after a translation of polygon F G H two units to the left and four units up? Answer 43 P a g e

44 9. Which transformations produce images that are congruent to their pre-image? 10. Which transformation does not produce an image that is congruent to its preimage? 11. a.) What are the two criteria for two figures to be similar? b.) What are the two criteria for two congruent figures? 12. Which transformations produce figures that have corresponding angles that are congruent? 13. The table below shows the coordinates of triangle RUN and the coordinates of R in triangle R U N. Triangle R U N is a dilation of triangle RUN. (4 pts) (8.G.1) Triangle RUN Triangle R U N R (6, 4) R (3, 2) U (-8, 0) U N (2, -2) N Part A Fill in the table above for the coordinates of point U and point N. Part B On the graph below, draw and label triangle R U N after a translation of R U N using the rule x, y x 2, y 5. Part C Which part(s) of the resulting figure are congruent to the original? 44 P a g e

45 14. John rotated the figure below 180 about the origin. (6pts) (8.G.4) John then dilated the figure with a scale factor of ½ with the center of dilation at the origin. a. Graph and label the final image on the grid provided. b. Are the image and the original figure congruent to each other, similar to each other, or neither? Explain how you determined your answer. 45 P a g e

46 15. Describe how you could move the solid polygon to exactly match the dashed polygon using a series of two transformations. (3pts) (8.G.4) 16. What sequence of transformations takes pentagon P to pentagon Q? 46 P a g e

47 17. Figure Q was the result of a sequence of transformations on Figure P both shown below. Which sequence of transformations could take Figure P to Figure Q? a. Reflection over the x-axis and translation 7 units right b. Reflection over the y-axis and translation 3 units down c. Translation 1 unit right and 180 o rotation about the origin d. Translation 4 units right and 180 o rotation about the origin 18. Figure L and Figure M are shown on the grid. Maria wants to transform figure L to figure M using only rotations, reflections, and translations. Which statement is true? a. The transformation can be done with a reflection followed by a rotation. b. The transformation can be done with a reflection followed by a translation. c. The transformation cannot be done because figure L is not congruent to figure M. d. The transformation cannot be done because figures L and M and are in different quadrants. 47 P a g e

48 19. Congruent rectangles HJKL and H J K L are shown on the coordinate grid below. Describe a sequence of transformations on rectangle HJKL that would result in rectangle H J K L. 20. A series of transformations on quadrilateral S resulted in quadrilateral T. The angle measures of quadrilateral T are congruent to those of quadrilateral S. The side lengths of quadrilateral T are twice as long as those quadrilateral S. Which transformation on quadrilateral S must be included to result in quadrilateral T? A dilation C rotation B reflection D translation 21. A triangle with vertices at A 1,1, B 2,1, and 1,4 image of vertex A has coordinates at (3, 1). C is translated. The Determine the coordinates of either the image of vertex B or the image of vertex C. Show your work. 48 P a g e

49 22. Rectangle JKLM is shown on the coordinate grid. Rectangle JKLM undergoes a sequence of transformations, resulting in rectangle J K L M. The length of the side K L is 6 units. The coordinates of vertex K are (-3,2), and the coordinates of vertex M are (3, -2). Describe a sequence of transformations to rectangle JKLM that would result in rectangle J K L M. Show your work. 23. Figure X and Figure Y are shown on the coordinate grid below. Which statement about figures X and Y must be true. A. A series of translations will transform figure X to figure Y, and the figures will be congruent. B. A 180⁰ clockwise rotation will transform figure X to figure Y, and the figures will be congruent. C. A series of translations will transform figure X to figure Y, but the figures will not be congruent. D. A 180⁰ clockwise rotation will transform figure X to figure Y, but the figures will not be congruent. 49 P a g e

50 24. When ABC was dilated by a scale factor of 2, centered at the origin, the result was its image A B C shown on the coordinate plane below. The vertices of A B C are A (-4, 4), B (-4, 6), and C (2, 4). What are the coordinates of the vertices of ABC? Vertices A(, ) B(, ) C(, ) Explain how you determined the coordinates of the vertices of ABC. Are ABC and A B C congruent to each other, similar to each other, or neither? Explain how you determined your answer. 50 P a g e

51 25. Quadrilateral ABCD is graphed on a coordinate plane. Abby reflected ABCD over the x-axis and then rotated it 90 clockwise about the origin. She labeled the final image EFGH. Manny dilated ABCD by a scale factor of 3 and then translated the resulting figure 2 units left. He labeled the final image PQRS. Identify a pair of quadrilaterals from the three quadrilaterals ABCD, EFGH, and PQRS that are congruent. Answer Identify a pair of quadrilaterals from the three quadrilaterals ABCD, EFGH, and PQRS that are similar but not congruent. Answer Describe a transformation on Abby s quadrilateral EFGH that would make the resulting image E F G H congruent to Manny s quadrilateral PQRS. 26. Rectangle FGHJ shown below, is translated 6 units right and 1 unit up to produce rectangle F G H J. Which statement about the side lengths of rectangle F G H J is true? A F G = 3 and G H = 5 B F G = 3 and G H = 6 C F G = 9 and G H = 5 D F G = 9 and G H = 6 51 P a g e

52 27. Mia enlarged a plan for an outdoor stage. The original plan is shown below. She dilated the outdoor stage by a scale factor of four with the center of dilation at the origin. Which ordered pair will be the coordinates of one of the new vertices? A (2,1) B (8,16) C (32, 4) D (32, 16) 28. Part A Fill in the table below for the coordinates of H, K, and L after a reflection over the x-axis. (4 pts)(8.g.3) Triangle HKL H (-2, 3) Triangle H K L H Part B: On the graph below, draw and label triangle HKL and triangle H K L. K (4, 2) K L (3, -2) L Part C: Explain how the coordinates of H changed to the coordinates of H. 52 P a g e

53 was translated using the rule 29. When ABC ( x, y) x 5, y 3, the result was its image. A B C shown on the coordinate plane below. The vertices of A B C are 4,4 A, B 4,6, and 2,4 C. (4pts) (8.G.2) What are the coordinates of the vertices of them. ABC? Explain how you determined What is the length of A C. How does this compare with the length of AC? Why? 30. Triangle M is similar to triangle N. Triangle M has two angles with measures of 32 and 93. Which two angle measures could be included in triangle N? A 32 and 58 B 32 and 74 C 93 and 55 D 93 and P a g e

54 1. a. In the figure below, L 1 is not parallel to L 2, and m is a transversal. Use patty paper to trace angles 1 4. Which, if any, are equal? Explain why. b. What other pairs of angles would be congruent based on question 1? c. Now translate angle 1-4 down to see if any of these angles are congruent to angles 5-8. What do you notice? d. Given that the m < 1 = 115 and the m < 2 = 65, what other angles can you determine? e. What do you notice about the sum of angle 1 and angle 2? f. What do you notice about the sum of angle 1, 2, 3, and 4? 54 P a g e

55 2. In the figure below, L 1 L 2, and m is a transversal. a. If the m < 1 = 40, what other angle measures do we know based on problem 1? b. Use patty paper to trace angles 1-4. What did you notice about the measures of 1 and 5? What transformation explains this? c. What do you notice about the measures of 3 and 7? What transformation explains this? Are there any other pairs of angles with this same relationship? If so, list them. Angles that are on the side of the transversal in the same position are called. If L 1 and L 2 are, then the corresponding angles are. 55 P a g e

56 d. What do you notice about the measures of 4 and 6? What transformations explain this? Is there another pair of angles with this same relationship? When angles are on sides of the transversal and inside the lines L 1 and L 2, they are called. If L 1 and L 2 are, then the alternate interior angles are. e. What do you notice about the measures of 2 and 8? What transformations explain this? Is there another pair of angles with this same relationship? When angles are on sides of the transversal and outside of the parallel lines, they are called angles. If L 1 and L 2 are, then the alternate exterior angles are. 56 P a g e

57 f. Based on the different relationships described, determine the measures of all the angles in the diagram. What do you notice about the measures of 4 and 5? What do you notice about the measures of 3 and 6? Angles that sum to are called angles. When angles are on the side of the transversal and inside of the parallel lines, they are called angles. If L 1 and L 2 are, then the same side interior angles are. 57 P a g e

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61 Exterior Angles of a Triangle 5. How can you calculate the measure of an exterior angle of a triangle? 6. How does that help you find the measure of the missing angle in the triangle below? 61 P a g e

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64 Practice: 1. In the diagram below, lines L1 and L2 are parallel. What are the measures of angles a g? 2. What are the measures of angles a and b in the triangle at the right? 3. What is the value of x in the diagram at the right? 4. What is the value of n in the diagram below? A 18 B 24 C 42 D P a g e

65 Find the value of the variable. Show your work or explain P a g e

66 1. a. In the following diagram m l cut by transversal t, find the measure of the other eight angles if m 4 = 110. m 1 = m 2 = m 3 = m 5 = m 6 = m 7 = m 8 = b. Identify a pair of corresponding angles: c. Identify a pair of alternate interior angles: d. Identify a pair of alternate exterior angles: 66 P a g e

67 2. In the diagram below lines l and k are parallel. Part A: What is the value of x? Show work. Part B: What is the measure, in degrees, of < ABC 67 P a g e

68 3. If m 3 = 90 and m CBD = 35. Determine the measures of the remaining angles in the diagram. Write the measures in the diagram. Explain how you determined each of them. Color congruent angles the same color. 4. Find the value of x. Show work. 5. Find the value of x. Show work and explain. 68 P a g e

69 6. If m 3 = 45 and m DCA = 135. Determine the measures of the remaining angles in the diagram. Write the measures in the diagram. Explain how you determined each of them. Color congruent angles the same color. 7. If l m. and m 1 = 50, determine the measures of the remaining angles. Give an example of alternate exterior angles. 69 P a g e

70 8. If l m. and m 2 = 110, determine the measures of the remaining angles. 9. The m 2 = 90. Find the measure of the remaining angles if m 4 = m 1 = m 3 = m 2 = m 5 = Tell how you know their measure without the use of a protractor. 10. Find the degree measure of YXZ. 70 P a g e

71 11. If the m 3 53 and m 4 85 find all the angles. Explain how you determined m In the following diagram a b.is cut by a transversal. The m 1 is 118 o. Find the m 4 and explain how you determined the measurement. 13. In parallelogram ABCD, the m A is 47 o. What is the measure of B? Explain. 71 P a g e

72 14. In ΔABC, the m B is twice the m A, and the m C is three times the m A. Find the number of degrees in each angle of the triangle. Show work. 15. Given rectangle ACDF, m BAD 20 and m CBD 60. Find the measurement of ADE. Explain how you arrived at the measurement for ADE. 16. If the m WXY 5x 40, m YXZ 70 and m XYZ 2x 14 find m XZY. 72 P a g e

73 17. In the following diagram m l cut by transversal t1 and t2. Find the measure of the angles 1, 3, & 5 and provide reasoning. Angle Measure Reason In the following diagram n1 n2.is cut by a transversal. The m 8 is 65 o. Find the m 2 and explain how you determined the measurement. 73 P a g e

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75 29. If the m 1 3x 7, m 11 x 17 and m find m In the following diagram x y.is cut by a transversal. The m 1 is 102 o. Find the m 3 and explain how you determined the measurement. 75 P a g e

76 31. Find the value of x. Show work and explain. 32. Find the m PRQ. Show work. 33. Two complementary angles measure 3x 5 and 5x 7. What is the measure of the larger angle? Show work. 34. Find the value of y. Find the degree measure of ABC. A D E B -6y+52 9y+7 C F 76 P a g e

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