1. Similar Triangles Similarity Laws Areas Irma Crespo 2010

2. Matching similar polygons scale factor ratio proportion triangles compares two quantities or measurements in fraction form ratio of the lengths of the corresponding sides of similar polygons ratio of the lengths of the corresponding sides of similar polygons congruent corresponding angles and proportionate ratio of corresponding side lengths polygon with three sides compares two equivalent ratios

3. Similar Triangles We Measured angle a = angle d = 70° angle c = angle f = 55° angle b = angle e = 55° = 55° 55° = 70° 3.75 inches 1.75 inches 3.25 inches 2 inches = 1.9 AB = AC = BC DE DF EF congruent corresponding angles and proportionate side lengths

4. Fast Facts triangle angle degreecongruent similar The sum of its interior angles is 180°.

5. Similarity Laws AA (Angle, Angle) Similarity SSS (Side, Side, Side) Similarity SAS (Side, Angle, Side) Similarity

6. AA Similarity If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Two corresponding angles are congruent. Two matching angles are congruent. J G I H L K

7. AA Similarity Examples Two angles of one triangle are congruent to two angles of another triangle. BC A E D F ADBFEC = = = SIMILAR

8. AA Similarity Example Two angles of one triangle are congruent to two angles of another triangle. AD BF EC = = = BC A E D F NOT SIMILAR

9. AA Similarity Example Two angles of one triangle are congruent to two angles of another triangle. AD BF EC = = = B C A These are right triangles. E DF SIMILAR

10. AA Similarity Example Two angles of one triangle are congruent to two angles of another triangle. AD BF EC = = = B C A 90° 60° E D F 90° 30° SIMILAR

11. SSS Similarity If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. There’s a scale factor. E D F A B C AB DE BC EF AC DF ==

12. SSS Similarity Example Corresponding side lengths are proportional. == AB DE BC EF AC DF A B C4 7 10 D E F 14 20 8 4848 = = 7 14 4848 1212 SIMILAR

13. SSS Similarity Example Corresponding side lengths are proportional. == AB DE AC DF BC EF 8484 = = 8484 5.2 2.8 BC A D EF 8 8 5.2 44 2.8 NOT SIMILAR

14. SAS Similarity If the measures of two sides of a triangle are proportional to the measures of corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. = AB DE AC DF A BC D E F and AD =

15. SAS Similarity If the measures of two sides of a triangle are proportional to the measures of corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. = AB DE AC DF and AD = A BC 10 12 65° D E F 56

16. SAS Similarity Example Two sides of a triangle are proportional to the measures of corresponding sides of another triangle and the included angles are congruent. = AB DE BC EF AD = and But are they included angles? NOT SIMILAR A BC 10 12 65° D E F 5 6

17. Your Turn Determine whether the triangles are similar. Justify your answer. = NM ZY ML YX = 1212 4 or 1 8 2 1 cm N LM 4 cm X Z Y 2 cm 8 cm MY = 90° = SIMILAR BY SAS

18. Your Turn Determine whether the triangles are similar. Justify your answer. AD = A BC E D F CE = SIMILAR BY AA

19. Areas of Similar Triangles Areas of similar triangles have a ratio that equals the square of the scale factor. Scale factor is 2.The square is 4. Scale factor is 1/3. Scale factor is 5. Scale factor is ¼. The square is 1/6. The square is 1/16. The square is 25. To square is to multiply the number by itself.

20. Finding Areas of Similar Triangles Triangle GHI and triangle JKL are similar with a scale factor of 3. If the area of triangle GHI is 81 square meters, find the are of triangle GHI. 10 cm H G I 7 cm 4 cm L K J 12 cm 21 cm 30 cm GHI JKL 9191 = 81 x 9x = 1*81 9x = 81 9 9 x = 9 cm

21. Find the Scale Factor of Two Areas The two triangles below are similar. The scale factor is the ratio of the corresponding sides. Find the scale factor of the two triangles. Then, find the scale factor of their areas. 35 in 28 in 21 in F E G 10 in 8 in 6 in X Y Z

22. Find the Scale Factor of Two Areas GEF XYZ 6 or 2 21 7 =(8*6) 35 in 28 in 21 in F E G 10 in 8 in 6 in X Y Z Find the scale factor. Find the scale factor of the areas. Use A = ½ *base*height 1212 = 24 in 2 = 1212 (28*21)=294 in 2 XYZ Area of GEF Area of Ratio of the areas. 24 or 4. 294 49

23. Summary Triangles are similar if they show AA Similarity, SSS Similarity, or SAS Similarity. The areas of similar triangles are the ratio that equals the square of the scale factor. To find the scale factors of two areas, just compute for the areas of each triangle and then, form the ratio of the areas.

24. Exit Slip Explain one of the Similarity Laws discussed today by either giving its meaning or making your own example or both. You will get an extra credit for this. Write your name and submit before the end of the hour.

25. Practice Worksheet Complete the practice worksheet. Work with a partner or on your own. Submit completed worksheet for grading. Solutions are discussed the next day. Exercises MI35 Lesson 6

26. Main Resources Lesson Plan Problems Math Connects: Concepts, Skills, and Problem Solving; Teacher Edition; Course 3, Volume 1 Columbus:McGraw-Hill, 2009. PowerPoint created by Irma Crespo. University of Michigan-Dearborn, School of Education. Winter 2010.