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Mastering Proportions: Solving for Missing Sides in Similar Figures

Dive into the world of similar figures and learn to solve problems involving missing lengths and angles using proportions. This lesson covers the essentials of identifying corresponding sides and angles in shapes like triangles and rectangles. Get familiar with the vocabulary of similar figures, understand how to express relationships through ratios, and sharpen your mathematical reasoning skills as per California State Standards. With practical examples and practice problems, you'll be well-equipped to tackle similar figure challenges confidently.

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Mastering Proportions: Solving for Missing Sides in Similar Figures

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  1. 8-86th grade math Similar Figures

  2. Objective • To use proportions to solve problems involving similar figures • Why? To know how to solve missing sides or lengths of triangles, rectangles, etc. that are similar in size.

  3. California State Standards NS 1.3 : Use proportions to solve problems (e.g… find the length of a side of a polygon similar to a known polygon) … MR 2.4: Use a variety of methods, such as words, numbers, symbols, … diagrams, … to explain mathematical reasoning.

  4. Vocabulary • Similar figures • Figures that have the same shape but not the same size (congruent = same size and shape) e f a b h g d c Rectangles: abcd ≈ efgh or abcd ~ efgh (similar to) ab ≈ ef ; cd ≈ gh; bc ≈ fg; da ≈ he (corresponding sides or lengths). Sides’ lengths change. ∠a ≈ ∠e; ∠b ≈ ∠f; ∠c ≈ ∠ g; ∠d ≈ ∠ h (corresponding angles). Angles’ measurement always stay the same, no matter how the angles change.

  5. How to Solve Missing Sides of Similar Figures 1) Locate corresponding sides. Color coding each corresponding side might help you to ‘see’ better. 2) Write a proportion. Corresponding sides make one ratio. (AC/DF) 3) Solve. * A short cut may be to ‘observe’ how the similar figure was ‘enlarged’ or ‘decreased’ and multiply or divide with that number. Triangles are similar. abc ~ def b e 6 4 3 2 d x f a 8 c ab ~ de ac df 6 ~ 3 8 X 24 ÷ 6 = 4 df = 4

  6. How to Solve Missing Angles of Similar Figures 1) Locate corresponding angles. 2) If finding a triangle’s missing angle, remember triangles angles add to 180°. If finding a rectangle’s missing angle, remember rectangles add to 360°. 3) Add the angles and subtract from either 180 (triangle) or 360 (rectangle). Triangles are similar. abc ~ def 103̊ 103̊ 30̊x ̊ 180 – (103 + 30) 180 – 133 = 47°

  7. Try It! Quadrilateral KLMN ~ to quadrilateral GHIJ. • m∠ K • HI • KN H I 110° L 2” M 9” 3” 110° K 70°N G 12” J 1) m∠ K = 360 - (110 + 110 + 70 + x) 360 – 270 = 70 2) HI = LM = KL 2 = 3 HI GH x 9 18 ÷ 3 = 6 HI = 6” 3) KN GJ = GH12 = 9 KN KL x 3 • ÷ 9 = 4 KN = 4”

  8. Objective Review • To use proportions to solve problems involving similar figures • Why? You now know how to solve missing sides or lengths of triangles, rectangles, etc. that are similar in size. • Use proportions to find missing sides of similar figures. • Write proportions with corresponding part of similar figures.

  9. Independent Practice • Complete problems 5-11 • Copy original problem first. • Show all work! • If time, complete Mixed Review: 12-18 • If still more time, work on Accelerated Math.

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