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Learn about the AA Similarity Postulate and how to use the SSS and SAS Similarity Theorems to determine if triangles are similar. Explains the concepts with examples and standardized test questions.

maryjohnston
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  1. Splash Screen

  2. Five-Minute Check (over Lesson 7–2) CCSS Then/Now Postulate 7.1: Angle-Angle (AA) Similarity Example 1: Use the AA Similarity Postulate Theorems Proof: Theorem 7.2 Example 2: Use the SSS and SAS Similarity Theorems Example 3: Standardized Test Example: Sufficient Conditions Theorem 7.4: Properties of Similarity Example 4: Parts of Similar Triangles Example 5: Real-World Example: Indirect Measurement Concept Summary: Triangle Similarity Lesson Menu

  3. Determine whether the triangles are similar. A. Yes, corresponding angles are congruent and corresponding sides are proportional. B. No, corresponding sides are not proportional. 5-Minute Check 1

  4. The quadrilaterals are similar. Find the scale factor of the larger quadrilateral to the smaller quadrilateral. A. 5:3 B. 4:3 C. 3:2 D. 2:1 5-Minute Check 2

  5. The triangles are similar.Find x and y. A.x = 5.5, y = 12.9 B.x = 8.5, y = 9.5 C.x = 5, y = 7.5 D.x = 9.5, y = 8.5 5-Minute Check 3

  6. 3 __ Two pentagons are similar with a scale factor of .The perimeter of the larger pentagon is 42 feet. What is the perimeter of the smaller pentagon? 7 A. 12 ft B. 14 ft C. 16 ft D. 18 ft 5-Minute Check 4

  7. 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. CCSS

  8. 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. Then/Now

  9. Concept

  10. Use the AA Similarity Postulate A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Example 1

  11. Since mB = mD, B D. Since mE = 80, A E. Use the AA Similarity Postulate By the Triangle Sum Theorem, 42 + 58 + mA = 180, so mA = 80. Answer:So, ΔABC ~ ΔEDF by the AA Similarity. Example 1

  12. Use the AA Similarity Postulate B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Example 1

  13. QXP NXM by the Vertical Angles Theorem. Since QP || MN, Q N. Use the AA Similarity Postulate Answer:So, ΔQXP ~ ΔNXM by AA Similarity. Example 1

  14. 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. Example 1

  15. 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. Example 1

  16. Concept

  17. Concept

  18. Use the SSS and SAS Similarity Theorems A.Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Answer:So, ΔABC ~ ΔDEC by the SSS Similarity Theorem. Example 2

  19. Use the SSS and SAS Similarity Theorems B.Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. By the Reflexive Property, M  M. Answer:Since the lengths of the sides that include M are proportional, ΔMNP ~ ΔMRS by the SAS Similarity Theorem. Example 2

  20. 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. Example 2

  21. 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 Example 2

  22. If ΔRST and ΔXYZ are two triangles such that = , which of the following would be sufficient to prove that the triangles are similar? A BC R  S D 2 __ RS ___ 3 XY Sufficient Conditions Example 3

  23. Read the Test Item You are given that = and asked to identify which additional information would be sufficient to prove that ΔRST ~ ΔXYZ. 2 __ RS ___ 3 XY Sufficient Conditions Example 3

  24. Solve the Test Item Since = , you know that these two sides are proportional with a scale factor of . Check each answer choice until you find one that supplies sufficient information to prove that ΔRST ~ ΔXYZ. 2 2 __ __ RS ___ 3 3 XY Sufficient Conditions Example 3

  25. Choice A If = , then you know that the other two sides are proportional. You do not, however, know whether the scale factor is , as determined by . Therefore, this is not sufficient information. 2 __ RT ST RS ___ ___ ___ 3 XZ YZ XY Sufficient Conditions Example 3

  26. Choice B If = = , then you know that all the sides are proportional with the same scale factor, . This is sufficient information by the SSS Similarity Theorem to determine that the triangles are similar. 2 __ RS RT RT ___ ___ ___ 3 XY XZ XZ Sufficient Conditions Answer: B Example 3

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

  28. Concept

  29. ALGEBRAGiven , RS = 4, RQ = x + 3, QT= 2x + 10, UT = 10, find RQ and QT. Parts of Similar Triangles Example 4

  30. Since because they are alternate interior angles. By AA Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar polygons, Parts of Similar Triangles Substitution Cross Products Property Example 4

  31. Parts of Similar Triangles Distributive Property Subtract 8x and 30 from each side. Divide each side by 2. Now find RQ and QT. Answer:RQ = 8; QT = 20 Example 4

  32. ALGEBRAGiven AB = 38.5, DE = 11, AC = 3x + 8, and CE =x + 2, find AC. A. 2 B. 4 C. 12 D. 14 Example 4

  33. Indirect Measurement 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 Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower? Understand Make a sketch of the situation. Example 5

  34. Indirect Measurement Plan In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent and that the two objects form the sides of two right triangles. Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate. So the following proportion can be written. Example 5

  35. Simplify. Divide each side by 2. Indirect Measurement Solve Substitute the known values and let x be the height of the Sears Tower. Substitution Cross Products Property Example 5

  36. 242 1452 CheckThe shadow length of the Sears Tower is or 121 times the shadow length of the light pole. Check to see that the height of the Sears Tower is 121 times the height of the light pole. = 121  ______ ______ 2 12 Indirect Measurement Answer: The Sears Tower is 1452 feet tall. Example 5

  37. LIGHTHOUSESOn her tripalong 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 feet6 inches. Jennie knows that her heightis 5 feet 6 inches. What is the height ofthe Cape Hatteras lighthouse to the nearest foot? A. 196 ft B. 39 ft C. 441 ft D. 89 ft Example 5

  38. Concept

  39. End of the Lesson

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