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Chapter 10 Review: Similarity

By Chris Lang and Mark Mshooshian. Chapter 10 Review: Similarity. Key Ideas…. Ratios/Proportions Geometric Mean Triangle Similarity Theorems Corresponding Altitudes Perimeter vs. Area of Similar Triangles Angle Bisector Theorem. Lesson 1: Ratios and Proportions. Ratio:

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Chapter 10 Review: Similarity

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  1. By Chris Lang and Mark Mshooshian Chapter 10 Review: Similarity

  2. Key Ideas… • Ratios/Proportions • Geometric Mean • Triangle Similarity Theorems • Corresponding Altitudes • Perimeter vs. Area of Similar Triangles • Angle Bisector Theorem

  3. Lesson 1: Ratios and Proportions Ratio: The number a to the number b is the number a/b Proportion: The equality between two ratios Geometric Mean: The number b is the geometric mean between the numbers a and c if a, b, and c are positive and if A B ________ = ________ B C

  4. Lesson 2: Similar Figures Similar: Figures are similar iff there is a correspondence between their vertices such that their corresponding sides are proportional and their corresponding angles are equal Magnitude: Relative size of the dilation image to the original Image in the form of a ratio

  5. Lesson 3: The Side-Splitter Theorem Side Splitter Theorem: If a line parallel to one side of a triangle intersects the other two sides in different points, it divides the sides in the same ratio Corollary to the Side Splitter Theorem: If a line parallel to one side of a triangle intersects the other two sides in different points, it cuts off segments proportional to the sides

  6. Lesson 4: The AA Similarity Theorem Theorem 45: The AA Theorem If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Corollary To AA Theorem: Two triangles similar to a third triangle are similar to each other. **You only need two angles to prove similarity because if you know two angles in a triangle, you automaticaly know the third by subtraction**

  7. Lesson 5: Proportions and Dilations Corresponding Altitudes • The altitudes of two similar triangles

  8. Lesson 5: Proportions and Dilations Theorem 46: Corresponding altitudes of similar triangles have the same ratio as that of the corresponding sides.

  9. Lesson 5: Proportions and Dilations The SAS Similarity Theorem: If an angle of one triangle is equal to an angle of another triangle and the sides including these angles are proportional, then the triangles are similar.

  10. Lesson 5: Proportions and Dilations The SSS Similarity Theorem: If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.

  11. Lesson 6: Perimeters and Areas of Similar Figures Perimeter vs. Area • Perimeter is measured in linear units • Area is measured in square units Theorem 47: The ratio of the perimeters of two similar polygons is equal to the ratio of the corresponding sides. Theorem 48: The ratio of the areas of two similar polygons is equal to the square of the ratio of the corresponding sides.

  12. Chapter 10 Extra Lesson: The Angle Bisector Theorem Angle Bisector Theorem: An angle bisector in a triangle divides the opposite side into segments that have the same ratio as the other two sides

  13. Chapter 10 Summary and Review Basic Ideas: • Dilation: center, magnitude pg 386 • Geometric mean pg 380 • Proportion pg 379 • Ratio pg 379 • Similar figures pg 386-387 • Similar triangles pg 386

  14. Theorems 44. The Side-Splitter Theorem: If a line parallel to one side of a triangle intersects the other two sides in different points it devides the sides in the same ratio. Corollary To Side Splitter: If a line parallel to one sideof a triangle intersects the other two sides in different points, it cuts off the segments proportional to the sides.

  15. Theorems 45. The AA Theorem: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Corollary To AA Theorem: Two triangles similar to a third triangle are similar to each other.

  16. Theorems 46. Corresponding altitudes of similar triangles have the same ratio as that of the corresponding sides. 47. The ratio of the perimeters of two similar polygons is equal to the ratio of the corresponding sides. 48.The ratio of the areas of two similar polygons is equal to the square of the ratio of the corresponding sides.

  17. Resulting Theorems The SAS Similarity Theorem: If an angle of one triangle is equal to an angle of another triangle and the sides including these angles are proportional, then the triangles are similar. The SSS Similarity Theorem: If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.

  18. Extra Theorem Angle Bisector Theorem: An angle bisector in a triangle that divides the opposite side into segments that have the same ratio as the other two sides.

  19. The End

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