1.  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.  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

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

3. Similar Polygons (7-2) Characteristics of similar polygons: 1.Corresponding angles are congruent (same shape) 2.Corresponding sides are proportional (lengths of sides have the same ratio) ABCD ~ EFGH Vertices must be listed in order when naming Characteristics of similar polygons: 1.Corresponding angles are congruent (same shape) 2.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. 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  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  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)  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  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  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  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.  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.  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.  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)  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:  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.  Find the geometric mean of 32 and 2.  Find the geometric mean of 6 and 20.

24. Similarity in Right Triangles (7-4)  

25.  

26.  

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