1. Chapter 7 Similarity

2. 7.1 Ratios and Proportions
A comparison of two quantities
A to B, A:B, or
Proportion
A Statement that two ratios are equal
Ex.

3. Properties of Proportions
If then 1 2 3 4

4. Cross-Product Property
The product of the extremes is equal to the product of the means
means
extremes

5. Examples
7.1 Examples

6. 7-2 Similar Polygons chapter 7.2 similarity.gsp
Two polygons are similar if
Corresponding angles are congruent
Corresponding sides are proportional
Similarity ratio
Ratio of the lengths of the corresponding sides
Practice with clickers

7. Golden Ratio

8. Golden Ratio
1:1.618
Also known as the golden rectangle

9. 7.2 Examples
Examples

10. 7.3 Proving Triangles Similar
AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

11. Similarity Theorems SAS Similarity SSS Similarity
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.
SSS Similarity
If the corresponding sides of two triangles are proportional, then the triangles are similar.

12. Examples
Examples

13. 7.4 Similarity in Right Triangles
Theorem 7-3
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.

14. Geometric Mean
The geometric mean of a and b is

15. Example
Find the geometric mean of 4 and 18

16. Corollaries to Theorem 7-3
Corollary 1
Piece of hypotenuse = Altitude
Altitude Other piece of hypotenuse
Corollary 2
Piece of hypotenuse = Leg
Leg Whole Hypotenuse

17. 7.5 Proportions in Triangles
Theorem 7-4 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. =

18. Corollary to Theorem 7-4
If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. c a d d b

19. Theorem 7-5 example 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. p m o n