Write and simplify ratios. Use proportions to solve problems.

1. Objectives Write and simplify ratios. Use proportions to solve problems.

2. Vocabulary ratio proportion extremes means cross products

3. A ratio compares two numbers by division. The ratio of two numbers a and b can be written as a to b, a:b, or , where b ≠ 0. For example, the ratios 1 to 2, 1:2, and all represent the same comparison.

4. Remember! In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.

5. Example 2: Using Ratios The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side? Let the side lengths be 4x, 7x, and 5x. Then 4x + 7x + 5x = 96 . After like terms are combined, 16x = 96. So x = 6. The length of the shortest side is 4x = 4(6) = 24 cm.

6. Check It Out! Example 2 The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle? x + y + z = 180° x + 6x + 13x = 180° 20x = 180° x = 9° y = 6x z = 13x y = 6(9°) z = 13(9°) y = 54° z = 117°

7. A proportion is an equation stating that two ratios are equal. In the proportion , the values a and d are the extremes. The values b and c are the means. When the proportion is written as a:b = c:d, the extremes are in the first and last positions. The means are in the two middle positions.

8. In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products.

9. The following table shows equivalent forms of the Cross Products Property.

10. Warm Up Solve each proportion. 1.2.3. x = 8 z = ±10

11. Figures that are similar(~) have the same shape but not necessarily the same size.

12. Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional.

13. Check It Out! Example 1 Identify the pairs of congruent angles and corresponding sides. B  G and C  H. By the Third Angles Theorem, A  J.

14. A similarity ratiois the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio of ∆ABC to ∆DEF is , or . The similarity ratio of ∆DEF to ∆ABC is , or 2.

15. Writing Math Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order.

16. Example 2B: Identifying Similar Polygons Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. ∆ABCD and ∆EFGH

17. Example 2B Continued Step 1 Identify pairs of congruent angles. P  R and S  W isos. ∆ Step 2 Compare corresponding angles. mW = mS = 62° mT = 180° – 2(62°) = 56° Since no pairs of angles are congruent, the triangles are not similar.

18. Check It Out! Example 2 Determine if ∆JLM ~∆NPS. If so, write the similarity ratio and a similarity statement. Step 1 Identify pairs of congruent angles. N  M, L  P, S  J

19. Thus the similarity ratio is , and ∆LMJ ~ ∆PNS. Check It Out! Example 2 Continued Step 2 Compare corresponding sides.

20. Helpful Hint When you work with proportions, be sure the ratios compare corresponding measures.

21. There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

24. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~.

25. Example 3: Finding Lengths in Similar Triangles Explain why ∆ABE ~ ∆ACD, and then find CD. Step 1 Prove triangles are similar. A A by Reflexive Property of , and B C since they are both right angles. Therefore ∆ABE ~ ∆ACD by AA ~.

26. Check It Out! Example 3 Explain why ∆RSV ~ ∆RTU and then find RT. Step 1 Prove triangles are similar. It is given that S T. R R by Reflexive Property of . Therefore ∆RSV ~ ∆RTU by AA ~.