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Warm Up #24

Learn about similar polygons and how to identify congruent angles and sides. Explore ratios and proportions in various scenarios and practice solving problems. Understand the concept of similarity in triangles and use proportions to find unknown measurements.

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Warm Up #24

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  1. Q  Z; R  Y; S  X; QR  ZY; RS  YX; QS  ZX Warm Up #24 1.If ∆QRS  ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides. Solve each proportion. 2.3. x = 9 x = 18

  2. homework • Ratio and Proportional • page 370_43, 45, 46, 47, 48, 50, 52, 62 • Page 376_13-16 and 21-28

  3. homework • Ratio and Proportional • page 370_ even • Page 376_13-16 and 21-28

  4. Ratios and Proportions

  5. There are many uses of ratios and proportions. We use them in map reading, making scale drawings and models, solving problems.

  6. The most recognizable use of ratios and proportions is drawing models and plans for construction. Scales must be used to approximate what the actual object will be like.

  7. A ratio is a comparison of two quantities by division. In the rectangles below, the ratio of shaded area to unshaded area is 1:2, 2:4, 3:6, and 4:8. All the rectangles have equivalent shaded areas. Ratios that make the same comparison are equivalent ratios.

  8. Ratio • A comparison of two quantities using division • 3 ways to write a ratio: • a to b • a : b • a/b

  9. Proportion • An equation stating that two ratios are equal • Example: • Cross products: means and extremes • Example: a and d = extremes b and c = means ad = bc

  10. There are 480 sophomores and 520 juniors in a high school. Find the ratio of juniors to sophomores. (ratio is when you simplify to the lowest value)

  11. Your Turn: solve these examples Ex: Ex:

  12. x = 9 x = 18

  13. Similar Polygons

  14. Q  Z; R  Y; S  X; QR  ZY; RS  YX; QS  ZX Congruent figures have the same size and shape. If ∆QRS  ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides.

  15. Similar Polygons • Similar polygons have: • Congruent corresponding angles • Proportional corresponding sides • Scale factor: the ratio of corresponding sides Polygon ABCDE ~ Polygon LMNOP A L B E M P Ex: N O C D

  16. For Polygons to be Similarcorresponding angles must be congruent, andcorresponding sides must be proportional(in other words the sides must have lengths that form equivalent ratios)

  17. Congruent figures have the same size and shape. Similar figures have the same shape but not necessarily the same size. The two figures below are similar. They have the same shape but not the same size.

  18. Let’s look at the two triangles we looked at earlier to see if they are similar.Are the corresponding angles in the two triangles congruent?Are the corresponding sides proportional? (Do they form equivalent ratios)

  19. Just as we solved for variables in earlier proportions, we can solve for variables to find unknown sides in similar figures.Set up the corresponding sides as a proportion and then solve for x. Ratios x/12 and 5/10 x5 1210 10x = 60 x = 6

  20. A.The two polygons are similar. Find x and y.

  21. If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.

  22. If LMNOP ~ VWXYZ, find the perimeter of each polygon.

  23. Lesson Quiz: Part I 1. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 2. The ratio of a model sailboat’s dimensions to the actual boat’s dimensions is . If the length of the model is 10 inches, what is the length of the actual sailboat in feet? no 25 ft

  24. Lesson Quiz: Part II 3. Tell whether the following statement is sometimes, always, or never true. Two equilateral triangles are similar. Always

  25. Similar Triangles

  26. Similar Triangles • Similar triangles have congruent corresponding angles and proportional corresponding sides Z Y A C X B angle A angle X angle B angle Y angle C angle Z ABC ~ XYZ

  27. Similar Triangles • Triangles are similar if you show: • Any 2 pairs of corresponding angles are congruent (AA Similarity) R B C T A S

  28. Similar Triangles • Triangles are similar if you show: • Any 2 pairs of corresponding sides are proportional and the included angles are congruent (SAS Similarity) R B 12 6 18 C T A 4 S

  29. 7.3: Similar Triangles • Triangles are similar if you show: • All 3 pairs of corresponding sides are proportional (SSS Similarity) R B 6 5 10 C 7 T 14 A 3 S

  30. Determine if the two triangles are similar.

  31. In the diagram we can use proportions to determine the height of the tree. 5/x = 8/288x = 140x = 17.5 ft

  32. The two windows below are similar. Find the unknown width of the larger window.

  33. These two buildings are similar. Find the height of the large building.

  34. Summary • Two polygons are similar if they have the same shape, but a different size. • If polygons are similar corresponding angles are congruent, and corresponding sides are proportional. • The ratio of any two corresponding sides is the scale factor. Geometry 8.3 Similar Polygons

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

  36. Determine whether the triangles are similar.

  37. A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

  38. B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

  39. A.Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

  40. B.Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

  41. A. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.

  42. B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.

  43. ALGEBRAGiven , RS = 4, RQ = x + 3, QT= 2x + 10, UT = 10, find RQ and QT.

  44. SKYSCRAPERSJosh 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 the Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?

  45. Questions • What is the golden rectangle? • Definition • Examples • What is the golden ratio? • Definition • examples

  46. Parallel Lines and Proportional Parts

  47. Parallel Lines and Proportional Parts • If a line is parallel to one side of a triangle and intersects the other two sides of the triangle, then it separates those sides into proportional parts. A Y X C B *If XY ll CB, then

  48. Parallel Lines and Proportional Parts • Triangle Midsegment Theorem • A midsegment of a triangle is parallel to one side of a triangle, and its length is half of the side that it is parallel to A E B *If E and B are the midpoints of AD and AC respectively, then EB = DC C D

  49. Parallel Lines and Proportional Parts • If 3 or more lines are parallel and intersect two transversals, then they cut the transversals into proportional parts C B A D E F

  50. Parallel Lines and Proportional Parts • If 3 or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal C B A D E If , then F

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