Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or

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1 Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning.

2 You solved problems by writing and solving equations. Write ratios. Write and solve proportions.

3 ratio extended ratios proportion extremes means cross products

4 Write and Simplify Ratios SCHOOL The number of students who participate in sports programs at Central High School is 520. The total number of students in the school is Find the athlete-to-student ratio to the nearest tenth.

5 The country with the longest school year is China, with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.) A. 0.3 B. 0.5 C. 0.7 D. 0.8

6 Use Extended Ratios In ΔEFG, the ratio of the measures of the angles is 5:12:13. Find the measures of the angles.

7 The ratios of the angles in ΔABC is 3:5:7. Find the measure of the angles. A. 30, 50, 70 B. 36, 60, 84 C. 45, 60, 75 D. 54, 90, 126

9 A. Use Cross Products to Solve Proportions

10 B. Use Cross Products to Solve Proportions

11 A. A. b = 0.65 B. b = 4.5 C. b = 14.5 D. b = 147

12 B. A. n = 9 B. n = 8.9 C. n = 3 D. n = 1.8

13 Use Proportions to Make Predictions PETS Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique s school, predict the total number of students with a dog or a cat.

14 Brittany randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job. A. 324 students B. 344 students C. 405 students D. 486 students

16 Content Standards G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Mathematical Practices 7 Look for and make use of structure. 3 Construct viable arguments and critique the reasoning of others.

17 You used proportions to solve problems. Use proportions to identify similar polygons. Solve problems using the properties of similar polygons.

18 similar polygons scale factor

20 Use a Similarity Statement If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.

21 If ΔGHK ~ ΔPQR, determine which of the following similarity statements is not true. A. HGK QPR B. C. K R D. GHK QPR

22 Identify Similar Polygons A. MENUS Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu:

23 Identify Similar Polygons B. MENUS Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu:

24 A. Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. A. BCDE ~ FGHI, scale factor = B. BCDE ~ FGHI, scale factor = C. BCDE ~ FGHI, scale factor = D. BCDE is not similar to FGHI Original: New:

25 B. Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. Original: New: A. BCDE ~ WXYZ, scale factor = B. BCDE ~ WXYZ, scale factor = C. BCDE ~ WXYZ, scale factor = D. BCDE is not similar to WXYZ

26 Use Similar Figures to Find Missing Measures A. The two polygons are similar. Find x.

27 Use Similar Figures to Find Missing Measures B. The two polygons are similar. Find y.

28 A. The two polygons are similar. Solve for a. A. a = 1.4 B. a = 3.75 C. a = 2.4 D. a = 2

29 B. The two polygons are similar. Solve for b. A. 1.2 B. 2.1 C. 7.2 D. 9.3

31 Use a Scale Factor to Find Perimeter If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.

32 If LMNOP ~ VWXYZ, find the perimeter of each polygon. A. LMNOP = 40, VWXYZ = 30 B. LMNOP = 32, VWXYZ = 24 C. LMNOP = 45, VWXYZ = 40 D. LMNOP = 60, VWXYZ = 45

33 Content Standards G.SRT.4 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 4 Model with mathematics. 7 Look for and make use of structure.

34 You used the AAS, SSS, and SAS Congruence Theorems to prove triangles congruent. Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems. Use similar triangles to solve problems.

36 Use the AA Similarity Postulate A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

37 Use the AA Similarity Postulate B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

38 A. Determine whether the triangles are similar. If so, write a similarity statement. A. Yes; ΔABC ~ ΔFGH B. Yes; ΔABC ~ ΔGFH C. Yes; ΔABC ~ ΔHFG D. No; the triangles are not similar.

39 B. Determine whether the triangles are similar. If so, write a similarity statement. A. Yes; ΔWVZ ~ ΔYVX B. Yes; ΔWVZ ~ ΔXVY C. Yes; ΔWVZ ~ ΔXYV D. No; the triangles are not similar.

42 Use the SSS and SAS Similarity Theorems A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

43 Use the SSS and SAS Similarity Theorems B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

44 A. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data. A. ΔPQR ~ ΔSTR by SSS Similarity Theorem B. ΔPQR ~ ΔSTR by SAS Similarity Theorem C. ΔPQR ~ ΔSTR by AA Similarity Theorem D. The triangles are not similar.

45 B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data. A. ΔAFE ~ ΔABC by SAS Similarity Theorem B. ΔAFE ~ ΔABC by SSS Similarity Theorem C. ΔAFE ~ ΔACB by SAS Similarity Theorem D. ΔAFE ~ ΔACB by SSS Similarity Theorem

46 Sufficient Conditions If ΔRST and ΔXYZ are two triangles such that = RS 2, which of the following would be sufficient XY 3 to prove that the triangles are similar? A B C R S D

47 Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar? A. AC = 4 DC 3 B. m A = 2m D C. AC = DC D. = BC DC BC EC 5 4

49 Parts of Similar Triangles ALGEBRA Given, RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT.

50 ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC. A. 2 B. 4 C. 12 D. 14

Indirect Measurement SKYSCRAPERS Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet.

51 Indirect Measurement SKYSCRAPERS Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet. Then he measured the length of Sears Tower s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?

52 LIGHTHOUSES On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot? A. 196 ft B. 39 ft C. 441 ft D. 89 ft

54 Content Standards G.SRT.4 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others.

55 You used proportions to solve problems between similar triangles. Use proportional parts within triangles. Use proportional parts with parallel lines.

56 midsegment of a triangle

58 Find the Length of a Side

59 A B C. 12 D

61 Determine if Lines are Parallel

62 A. yes B. no C. cannot be determined

64 Use the Triangle Midsegment Theorem A. In the figure, DE and EF are midsegments of ΔABC. Find AB.

65 Use the Triangle Midsegment Theorem B. In the figure, DE and EF are midsegments of ΔABC. Find FE.

66 Use the Triangle Midsegment Theorem C. In the figure, DE and EF are midsegments of ΔABC. Find m AFE.

67 A. In the figure, DE and DF are midsegments of ΔABC. Find BC. A. 8 B. 15 C. 16 D. 30

68 B. In the figure, DE and DF are midsegments of ΔABC. Find DE. A. 7.5 B. 8 C. 15 D. 16

69 C. In the figure, DE and DF are midsegments of ΔABC. Find m AFD. A. 48 B. 58 C. 110 D. 122

71 Use Proportional Segments of Transversals MAPS In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in between city blocks. Find x.

72 In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in between city blocks. Find x. A. 4 B. 5 C. 6 D. 7

74 Use Congruent Segments of Transversals ALGEBRA Find x and y.

75 Find a and b. A. 2 ; 3 B. 1; 2 C. 11; 3 2 D. 7; 3

76 Content Standards G.SRT.4 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others.

77 You learned that corresponding sides of similar polygons are proportional. Recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles. Use the Triangle Bisector Theorem.

80 In the figure, ΔLJK ~ ΔSQR. Find the value of x. Use Special Segments in Similar Triangles

81 In the figure, ΔABC ~ ΔFGH. Find the value of x. A. 7 B. 14 C. 18 D. 31.5

82 Use Similar Triangles to Solve Problems ESTIMATING DISTANCE Sanjay s arm is about 9 times longer than the distance between his eyes. He sights a statue across the park that is 10 feet wide. If the statue appears to move 4 widths when he switches eyes, estimate the distance from Sanjay s thumb to the statue.

83 Use the information from Example 2. Suppose Sanjay turns around and sees a sailboat in the lake that is 12 feet wide. If the sailboat appears to move 4 widths when he switches eyes, estimate the distance from Sanjay s thumb to the sailboat. A. 324 feet B. 432 feet C. 448 feet D. 512 feet

85 Find x. Use the Triangle Angle Bisector Theorem

86 Find n. A. 10 B. 15 C. 20 D. 25

87 Content Standards G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 6 Attend to precision. 4 Model with mathematics.

88 You identified congruence transformations. Identify similarity transformations. Verify similarity after a similarity transformation.

89 dilation similarity transformation center of dilation scale factor of a dilation enlargement reduction

91 Identify a Dilation and Find Its Scale Factor A. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

92 Identify a Dilation and Find Its Scale Factor B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation.

93 A. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. A. reduction; 1 2 B. reduction; 1 3 C. enlargement; 2 D. enlargement; 3

94 B. Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. A. reduction; B. reduction; C. enlargement; D. enlargement; 2 3 2

95 Find and Use a Scale Factor PHOTOCOPYING A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image?

96 PHOTOGRAPHS Mariano wants to enlarge a picture he took that is 4 inches by 7.5 inches. He wants it to fit perfectly into a frame that is 400% of the original size. What will be the dimensions of the enlarged photo? A. 15 inches by 25 inches B. 8 inches by 15 inches C. 12 inches by 22.5 inches D. 16 inches by 30 inches

97 Verify Similarity after a Dilation A. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: M( 6, 3), N(6, 3), O( 6, 6) image: D( 2, 1), F(2, 1), G( 2, 2)

98 Verify Similarity after a Dilation B. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: G(2, 1), H(4, 1), I(2, 0), J(4, 0) image: Q(4, 2), R(8, 2), S(4, 0), T(8, 0)

99 A. Graph the original figure and its dilated image. Then determine the scale factor of the dilation. original: B( 7, 2), A(5, 2), D( 7, 7) image: J( 3, 0), K(1, 0), L( 3, 3) A. B. C. D

100 B. Graph the original figure and its dilated image. Then determine the scale factor of the dilation. original: A(4, 3), B(6, 3), C(4, 2), D(6, 2) image: E(6, 4), F(10, 4), G(6, 2), H(10, 2) A. 2 B. 1 3 C. 3 D. 4

101 Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 4 Model with mathematics. 7 Look for and make use of structure.

102 You used scale factors to solve problems with similar polygons. Interpret scale models. Use scale factors to solve problems.

103 scale model scale drawing scale

104 Use a Scale Drawing MAPS The distance between Boston and Chicago on a map is 9 inches. If the scale of the map is 1 inch: 95 miles, what is the actual distance from Boston to Chicago?

105 MAPS The distance between Cheyenne, WY, and Tulsa, OK, on a map is 8 inches. If the scale of the map is 1 inch : 90 miles, what is the actual distance from Cheyenne to Tulsa? A. 800 miles B. 900 miles C. 630 miles D. 720 miles

106 Find the Scale A. SCALE MODEL A miniature replica of a fighter jet is 4 inches long. The actual length of the jet is 12.8 yards. What is the scale of the model?

107 Find the Scale B. SCALE MODEL A miniature replica of a fighter jet is 4 inches long. The actual length of the jet is 12.8 yards. How many times as long as the actual is the model jet?

108 A. SCALE MODEL A miniature replica of a fire engine is 9 inches long. The actual length of the fire engine is 13.5 yards. What is the scale of the replica? A. 2 in. : 3 yd B. 1 in. : 3 yd C. 2 in. : 5 yd D. 3 in. : 4 yd

109 B. SCALE MODEL A miniature replica of a fire engine is 9 inches long. The actual length of the fire engine is 13.5 yards. How many times as long as the model is the actual fire engine? A. 48 B. 54 C. 60 D. 63

110 Construct a Scale Model SCALE DRAWING Gerrard is making a scale model of his classroom on an 11-by-17 inch sheet of paper. If the classroom is 20 feet by 32 feet, choose an appropriate scale for the drawing and determine the drawing s dimensions.

111 ARCHITECTURE Alaina is an architect making a scale model of a house in a 15-by-26 inch display. If the house is 84 feet by 144 feet, what would be the dimensions of the model using a scale of 1 in. : 6 ft? A. 14 in. 24 in. B. 14 in. 25 in. C. 15 in. 25 in. D. 15 in. 26 in.

Section 6 1: Proportions Notes

Date: Section 6 1: Proportions Notes Write Ratios: Ratio: Ways to express the ratio a to b: Example #1: The total number of students who participate in sports programs at Woodland Hills High School is