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Similarity

Similarity. When two objects are congruent , they have the same shape and size. Two objects are similar if they have the same shape, but different sizes. Their corresponding parts are all proportional . Any kind of polygon can have two that are similar to each other. Similarity.

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Similarity

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  1. Similarity • When two objects are congruent, they have the same shape and size. • Two objects are similar if they have the same shape, but different sizes. • Their corresponding parts are all proportional. • Any kind of polygon can have two that are similar to each other.

  2. Similarity • Examples: 2 squares that have different lengths of sides. • 2 regular hexagons

  3. Similar Polygons (7-2) Characteristics of similar polygons: • Corresponding angles are congruent (same shape) • Corresponding sides are proportional (lengths of sides have the same ratio) ABCD ~ EFGH Vertices must be listed in order when naming

  4. Similar Polygons (7-2) ABCD ~ EFGH Complete the statements.

  5. Similar Polygons (7-2) • Determine whether the parallelograms are similar. Explain.

  6. Similar Polygons (7-2) • Scale factor- ratio of the lengths of two corresponding sides of two similar polygons • The scale factor can be used to determine unknown lengths of sides

  7. Similar Polygons (7-2) • If ABC ~ YXZ, find the scale factor of the large triangle to the small and find the value of x. scale factor = 5/2 x= 16

  8. Example from Similar Polygons Worksheet • Are the two polygons shown similar? • Corresponding angles must be congruent • All pairs of corresponding sides must be proportional (same scale factor)

  9. Example from Using Similar Polygons Worksheet • Given two similar polygons. Find the missing side length. • Redraw one of the polygons so corresponding sides match up (if needed) • Determine the scale factor • Set up a proportion and solve for the missing side length

  10. Similar Polygons (7-2) • Homework • Similar Polygons worksheet #1-17 odd • Using Similar Polygons worksheet #1-15 odd

  11. Scale Drawing • Problem 2 on p.443 • Complete Similarity Application Problems • More practice p.444 #9, 13, 15, 19, 23, and 25

  12. Similar Triangles (7-3) • AA ~ Postulate – If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

  13. Similar Triangles (7-3) • SAS ~ Theorem – If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.

  14. Similar Triangles (7-3) • SSS ~ Theorem – If the corresponding sides of two triangles are proportional, then the triangles are similar.

  15. Similar Triangles (7-3)

  16. Similar Triangles (7-3) • Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain. • No the vertical angle is not between the two pairs of proportional sides.

  17. Similar Triangles (7-3)

  18. Similar Triangles (7-3) • Find the value of x.

  19. Indirect Measurement (7-3) • When a 6 ft man casts a shadow 18 ft long, a nearby tree casts a shadow 93 ft long. How tall is the tree?

  20. Homework • 7-4 A Postulate for Similar Triangles (AA) worksheet #1-12 all • 7-5 Theorems For Similar Triangles (SSS and SAS) worksheet #1-6 all • Similar Triangles Worksheet (all three methods)

  21. Similarity in Right Triangles (7-4) • Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

  22. Similarity in Right Triangles (7-4) • Geometric mean For any two positive numbers a and b, x is the geometric mean if Another way to find the geometric mean:

  23. Similarity in Right Triangles (7-4) • Find the geometric mean of 32 and 2. • Find the geometric mean of 6 and 20.

  24. Similarity in Right Triangles (7-4)

  25. Similarity in Right Triangles (7-4)

  26. Similarity in Right Triangles (7-4)

  27. Homework • 8-1 worksheet #24-31 all

  28. Proportions in Triangles (7-5) • Side-Splitter Theorem – If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

  29. Proportions in Triangles (7-5) • Solve for x. • x = 9

  30. Proportions in Triangles (7-5) • Corollary: If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

  31. Proportions in Triangles (7-5) • Solve for x. • x = 24

  32. Proportions in Triangles (7-5) • Triangle-Angle-Bisector Theorem – If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. • x = 18

  33. Proportions in Triangles (7-5) • Solve for x. • x = 11.25

  34. Homework • 7-6 Proportional Lengths worksheet • Proportional Parts in Triangles and Parallel Lines worksheet • p.475 #9-12, 15-22 • Study for test

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