Similar Triangles

1. Similar Triangles Similarity Laws Areas Irma Crespo 2010

2. 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 congruent corresponding angles and proportionate ratio of corresponding side lengths polygon with three sides compares two equivalent ratios Matching

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

4. triangle angle degree congruent similar Fast Facts 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. H K J L I G 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.

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

8. = E A E F C D A B = = F D B C AA Similarity Example • Two angles of one triangle are congruent to two angles of another triangle. NOT SIMILAR

9. C = E A D B F E C = = D F A B AA Similarity Example • Two angles of one triangle are congruent to two angles of another triangle. SIMILAR These are right triangles.

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

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

12. A 7 4 C 10 B E 20 AB DE BC EF AC DF 8 F 14 D SSS Similarity Example • Corresponding side lengths are proportional. = = 4 8 7 14 4 8 = = 1 2 SIMILAR

13. A D 4 4 8 8 AB DE AC DF BC EF E F 2.8 B C 5.2 SSS Similarity Example • Corresponding side lengths are proportional. = = 8 4 8 4 5.2 2.8 = = NOT SIMILAR

14. = D A D A AB DE AC DF F E B C 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. = and

15. = D A D A 5 65° 6 AB DE AC DF 65° 12 10 F E B C 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. = and

16. = D D A 5 65° A AB DE 65° BC EF 10 6 F E 12 B C 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. But are they included angles? = and NOT SIMILAR

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

18. = = A A D C E B C D F E Your Turn • Determine whether the triangles are similar. Justify your answer. 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. The square is 1/6. Scale factor is 1/3. The square is 1/16. Scale factor is ¼. The square is 25. Scale factor is 5. To square is to multiply the number by itself.

20. JKL GHI I 10 cm K 7 cm G H 4 cm 30 cm 21 cm J L 12 cm 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. 9 1 81 x = 9x = 1*81 9x = 81 9x = 81 9 9 x = 9 cm

21. Y E 10 in 6 in Z X 8 in 35 in 21 in G F 28 in 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.

22. GEF XYZ GEF XYZ Y Area of Area of E 10 in 6 in Z X 8 in 35 in 21 in G F 28 in Find the Scale Factor of Two Areas Find the scale factor. 6 or 2 21 7 Find the scale factor of the areas. Use A = ½ *base*height 1 2 24 in2 = (8*6) = 1 2 = (28*21) = 294 in2 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. Complete the practice worksheet. Exercises MI35 Lesson 6 Work with a partner or on your own. Submit completed worksheet for grading. Solutions are discussed the next day. Practice Worksheet

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.