Understanding Similar Triangles in Geometry

1. G B O B 3 F x 15 R H 7 A C J P H 10 +2 +2 +1 +1 +1 +1 +1 +1 Have out: Pencil, highlighter, red pen, GP NB, textbook, calculator, HW U8D4 Bellwork: Solve for x. 1. ∆JOR~∆PHB 2. ∆ABC~∆GFH 4 3 7 6 4+x

2. x 4 6 5 4 6 Add to your notes... We have determined that if triangles are similar, then the ratios of their corresponding sides are the same. This allows us to write PROPORTIONS to find unknown sides in similar triangles. Sometimes you may have overlapping triangles. Example: A First establish similar s: O Write a proportion: (given) D D (reflexive property) G T DOG ~ DAT (AA ~ Thm) or It is important to remember to use actual SIDES of triangles, NOT PIECES of sides!

3. m A E D 1 2 B C (// corresponding s ) S -34 Eleanor and John know that the line m is parallel to . They want to show that: Hold on a minute! What angles are equal? This is easy. We have parallel lines so it is similar by AA Theorem. • Are there any pairs of angles that we can conclude are equal? Justify your answer. • Are the triangles (ΔAED and ΔABC) similar? Explain. B 1; C  2 (Or use one of the above and A  A by Reflexive Property.) Yes ∆AED ~∆ABC, by AA~ Theorem.

4. A 3 x E D B C c) When John sees that the triangles are similar he suggests redrawing them as shown. 3 x Look, now you can just write an PROPORTION like we did in some earlier problems. + + 8 5 Solve for x. or

5. d) Solve for x. A 4 6 E D What would happen if x was the length of a lower piece of the large triangle? A 6 4 + + x 7 B C or

6. A 6 8 E D A e) In this figure suppose AE = 6, AD = 8, and DC = 16. Draw figures, write an equation, and solve for EB. 6 8 + + x 16 B C

7. Quiz Time Clear your desk except for a pencil, highlighter, and a calculator! After the quiz, work on the rest of the assignment S 35-43!

8. Solve for x in each figure. Be consistent in matching corresponding parts of similar figures. (Don’t forget to justify!) S -35 A b) a) A 8 6 4 R B 5 14 6 E 3 C F x S E 8 x D d) S c) O x 6 x C D R 22 8 x N E A 32 E O 13

9. Solve for x in each figure. Be consistent in matching corresponding parts of similar figures. (Don’t forget to justify!) S -35 A a) 8 4 B 5 6 E 3 C x D (Given) (reflexive property) AEB ~ ADC (AA  Thm)

10. b) A 6 R 14 F x S E 8 (Given) (reflexive property) REF ~ ASF (AA  Thm)

11. c) O x C 22 N E A 32 13 (Given) (reflexive property) ACE ~ NOE (AA  Thm)

12. d) S x 6 D R 8 x E O (Given) (reflexive property) DRS ~ EOS (AA  Thm) or

13. To find the distance AB across the swamp, sights were taken as shown with parallel to . S -36 a) Name the similar triangles, and justify. Alternate Interior  Thm. A B Alternate Interior  Thm. AA ~ Theorem 218 m 396 m b) Which side corresponds to ? E c) With these measures, find AB: 95 m C D 176 m

14. 1 2 2 3 3 4 S -37 Shalley starts with a 1x2 rectangle. She is going to draw a sequence of 100 different rectangles by taking her original 1x2 rectangle and adding one to both the length and the width. First rectangle Second rectangle Third rectangle • What are the dimensions of the 15th rectangle? 15x16 • Are all the rectangles similar? If not. Explain why not. If so what is the ratio of similarity. No! If you take the side ratios of the third and second triangles, then you get two different ratios. To have similar rectangles, the ratios must be the same.

15. S -38 The figure at the right has a smaller isosceles triangle (ΔPRS) drawn inside a larger isosceles triangle (ΔPQT). Both triangles have a common vertex (P). Is It true that ΔPQS is congruent to ΔPTR? P Statement Reason Q R S T 1) Definition of isosceles (given) 1) 2) 2) Isosceles  Theorem 3) Definition of isosceles (given) 3) 4) Isosceles  Theorem 4) 5) 5) AAS  ASA

16. Keep working on the rest of the assignment S 39 – 43.