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Ratio

A. 8. 10. B. C. 6. Ratio. A ratio is a comparison of two numbers such as a : b. . Ratio:. When writing a ratio, always express it in simplest form. Example:. Now try to reduce the fraction. . Proportion. Proportion: An equation that states that two ratios are equal. Terms.

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Ratio

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  1. A 8 10 B C 6 Ratio A ratio is a comparison of two numbers such as a : b. Ratio: When writing a ratio, always express it in simplest form. Example: Now try to reduce the fraction.

  2. Proportion Proportion: An equation that states that two ratios are equal. Terms First Term Third Term Second Term Fourth Term To solve a proportion, cross multiply the proportion

  3. Practice Reduce the following fractions. 1) 2) 3) Solve the following equations. What is the method? 4) 5) 6)

  4. Practice worksheet • On my website • www.mshalllovesmath.wordpress.com • Period 2 • Proportions

  5. Standard • 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.

  6. Objective • Using prior knowledge of proportions and angles and given knowledge about similar triangles students will determine if triangles are similar.

  7. Similar Figures Definition: Two figures are similar if • their corresponding sides are proportional, and • their corresponding angles are equal. Similar figureshave the same shape but possibly different sizes. Similarity Statement: ΔABC ~ ΔDEF Notes: ~ => “is similar to” The order of the letters matters! A D E F B C

  8. Match pairs of similar shapes.

  9. 1) Corresponding angles are congruent: 2) Corresponding sides are proportional: A D E F B C = = The fractions all reduce to a common similarity ratio.

  10. Example 1 Identify the pairs of congruent angles and corresponding sides. 0.5 N  Q and P  R, M  T

  11. Example 2 Identify the pairs of congruent angles and corresponding sides. B  G and C  H, A  J.

  12. Ex 3. Identify the pairs of congruent angles and corresponding sides. 85° 14 2 85° 45° 50° 45° 50° Are the triangles similar? How do you know?

  13. Identify the pairs of congruent angles and corresponding sides. Congruent Angles N  M, L  P, S  J

  14. 85° 85° 45° 50° 45° 50° Identify the pairs of congruent angles and corresponding sides.

  15. Polygon Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. A  E, B  F, C  G, and D  H.

  16. Ratios and Proportion worksheets and similar figures and proportions Picture frame problem

  17. Objective: Students will use their prior knowledge on proportions, and similarity to solve for a missing side.

  18. Example 1 ∆SUT ~ ∆SRQ. Use a proportion to solve for x. S 7 R S 18 21 x Q U T

  19. A Example 2: Set up a proportion and find x. ΔRDE ~ ΔAML R 9 L x 8 12 E D M

  20. Example 3 • ∆SUT ~ ∆SRQ. Use a proportion to solve for x. 3 x 3 5 x

  21. Example 4 • ∆PST~ ∆PQR • Solve for x. x 4 12 6

  22. More practice with proportions and similar figures • Go to www.mshalllovesmath.wordpress.com • Click on your period and open the link More practice with proportions and similar figures Work on it on paper or on adobe reader

  23. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Objective: Students will investigate AA criterion and SSS criterion for triangle similarity and be able to use the criterion to identify similar triangles

  24. Side-Side-Side (SSS~) M Q c a d f P b R L N e If ___ = ___ = ___, then the triangles are similar. Note: If sides are proportional, then corresponding angles must be congruent, and the Δs ~.

  25. W Example 1: Are the two triangles similar? Why or why not? Yes SSS~ What is the similarity statement? Δ_ZHT______ ~ Δ_WKR_______ If m∠T = 39° and m∠W = 57°, find the measures of the remaining angles in both triangles. Z Are they giving us side lengths or angle measures? 12 9 3 4 H T 6 R K 8

  26. W Example 2: Are the two triangles similar? Why or why not? Yes SSS~ What is the similarity statement? Δ__ZHT_____ ~ Δ_WRT_______ If m∠T = 41° and m∠R = 67°, find the measures of the remaining angles in both triangles. Z Are they giving us side lengths or angle measures? 9 5 18 10 H T 8 R K 16

  27. Angle-Angle (AA ~) M Q a c d f b R P L N e 2 If _____ pairs of ________________ angles are ______________, then the triangles are similar. corresponding congruent Note: If ∠____ ≅ ∠ ____ and ∠ ____≅ ∠____, then the corresponding sides must be proportional and the Δ’s ~..

  28. Are they giving us side lengths or angle measures? B Example Are the two triangles similar? Why or why not? Yes AA~ What is the similarity statement? Δ_______ ~ Δ ________ 53° C Are they giving us side lengths or angle measures? S 12 61 ° 53° 61° F E Q

  29. Focus Question Why do we only need to show that TWO pairs of corresponding angles are congruent to prove similarity? Think: How did you find the missing angle in BOTH triangles? B 53° C S 61 ° 53° 61° F E Q

  30. Example: Are the two triangles similar? Why or why not? No the triangles do not Have 2 corresponding congruent angles 35° 58°

  31. Classwork #3 • See Worksheet “CW #3 SSS~ and AA~”

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