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Similar and Congruent Solids: Identifying, Comparing, and Finding Unknown Values

This lesson covers the concepts of similar and congruent solids, including identifying, comparing, and finding unknown values using ratios. Students will learn how to determine if solids are similar or congruent, find the scale factor, and solve for unknown measurements.

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Similar and Congruent Solids: Identifying, Comparing, and Finding Unknown Values

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 12–7) Then/Now New Vocabulary Key Concepts: Similar Solids Key Concepts: Congruent Solids Example 1: Identify Similar and Congruent Solids Theorem 12.1 Example 2: Use Similar Solids to Write Ratios Example 3: Real-World Example: Use Similar Solids to Find Unknown Values Lesson Menu

  3. A. B. C. D. Name a line not containing point P on the sphere. 5-Minute Check 1

  4. Name a triangle in the sphere. A.ΔVQS B.ΔRTU C.ΔPQR D.ΔPXW 5-Minute Check 2

  5. A. B. C. D. TU Name a segment containing point Q in the sphere. 5-Minute Check 3

  6. Tell whether the following statement from Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain why.If B is between A and C, then AB + BC = AC. A. Yes, through 2 points there is exactly one line. B. Yes, the points on any great circle or arc of a great circle can be put into one to one correspondence with real numbers. C. No, AC may not be the distance from A to C through B. It may be the distance the other direction around the sphere. 5-Minute Check 4

  7. Which of the following is represented by a line in spherical geometry? A. triangle B. great circle C. radius D. diameter 5-Minute Check 5

  8. You compared surface areas and volumes of spheres. • Identify congruent or similar solids. • Use properties of similar solids. Then/Now

  9. similar solids • congruent solids Vocabulary

  10. Concept

  11. Concept

  12. Simplify. Identify Similar and Congruent Solids A. Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. Find the ratios between the corresponding parts of the square pyramids. Substitution Example 1

  13. Simplify. Simplify. Identify Similar and Congruent Solids Substitution Substitution Answer: The ratios of the measures are equal, so we can conclude that the pyramids are similar. Since the scale factor is not 1, the solids are not congruent. Example 1

  14. Identify Similar and Congruent Solids B. Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. Compare the ratios between the corresponding parts of the cones. Example 1

  15. Identify Similar and Congruent Solids Substitution Simplify. Substitution Answer: Since the ratios are not the same, the cones are neither similar nor congruent. Example 1

  16. A. Determine whether the pair of solids is similar, congruent, or neither. A. similar B. congruent C. neither Example 1

  17. B. Determine whether the pair of solids is similar, congruent, or neither. A. similar B. congruent C. neither Example 1

  18. Concept

  19. Use Similar Solids to Write Ratios Two similar cones have radii of 9 inches and 12 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone? First, find the scale factor. Write a ratio comparing the radii. Example 2

  20. If the scale factor is , then the ratio of the volumes is . Use Similar Solids to Write Ratios Answer: So, the ratio of the volume is 27:64. Example 2

  21. Two similar cones have radii of 5 inches and 15 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone? A. 1:3 B. 1:9 C. 1:27 D. 1:81 Example 2

  22. Use Similar Solids to Find Unknown Values SOFTBALLS The softballs shown are similar spheres. Find the radius of the smaller softball if the radius of the larger one is about 1.9 cubic inches. UnderstandYou know the volume of the softballs. PlanUse Theorem 12.1 to write a ratio comparing the volumes. Then find the scale factor and use it to find r. Example 3

  23. = ≈ Write as . Use Similar Solids to Find Unknown Values Solve Write a ratio comparing volumes. Simplify. Example 3

  24. Use Similar Solids to Find Unknown Values Ratio of radii Scale factor Find the cross products. r≈ 1.45 Solve for r. Answer: So, the radius of the smaller softball is about 1.45 inches. Example 3

  25. Use Similar Solids to Find Unknown Values Check  Example 3

  26. CONTAINERS The containers below are similar cylinders. Find the height h of the smaller container. A. 2 in. B. 3 in. C. 4 in. D. 5 in. Example 3

  27. End of the Lesson

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