Similar Triangles Chapter 7-3 TARGETS Identify similar triangles. Use similar triangles to solve problems.

Content Standards G-SRT.2 and G-SRT.3 Understand similarity in terms of similarity transformations G-SRT.4 and G-SRT.5 Prove theorems using similarity Mathematical Practices 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively. 4 Model with mathematics 6 Attend to precision. Lesson 3 MI/Vocab

Triangle Similarity is: Lesson 3 TH2

Writing Proportionality Statements Given BTW ~ ETC Write the Statement of Proportionality Find mTEC Find TE and BE T W B C E 34o 3 20 mTEC = mTBW = 79o 79o 12

AA  Similarity Theorem If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. If K  Y and J  X, then  JKL ~  XYZ. K J L Y X Z

Example Are these two triangles similar? Why? N M P Q R S T

SSS  Similarity Theorem If the corresponding sides of two triangles are proportional, then the two triangles are similar. B A C Q P R

Which of the following three triangles are similar? J H G 6 14 10 F E D 6 8 4 A C B 12 6 9 ABC and FDE? ABC~ FDE SSS ~ Thm Scale Factor = 3:2 Longest Sides Shortest Sides Remaining Sides

Which of the following three triangles are similar? J H G 6 14 10 F E D 6 8 4 A C B 12 6 9 ABC and GHJ ABC is not similar to DEF Longest Sides Shortest Sides Remaining Sides

SAS  Similarity Theorem If one angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. K J L Y X Z ass Pantograph

Prove RTS ~ PSQ S  S (reflexive prop.) SPQ  SRT SAS  ~ Thm. S 5 15 12 4 SPQ  SRT SAS  ~ Thm.

Are the two triangles similar? P Q R T 10 15 12 9 NQP  TQR Not Similar

How far is it across the river? 2 yds 5 yds x yards 42 yds 2x = 210 x = 105 yds

Are Triangles Similar? In the figure, , and ABC and DCB are right angles. Determine which triangles in the figure are similar. Lesson 3 Ex1

by the Alternate Interior Angles Theorem. Are Triangles Similar? by the Alternate Interior Angles Theorem. Vertical angles are congruent, Answer: Therefore, by the AA Similarity Theorem, ΔABE ~ ΔCDE. Lesson 3 Ex1

In the figure, OW = 7, BW = 9, WT = 17. 5, and WI = 22. 5 In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5. Determine which triangles in the figure are similar. A. ΔOBW ~ ΔITW B. ΔOBW ~ ΔWIT C. ΔBOW ~ ΔTIW D. ΔBOW ~ ΔITW Lesson 3 CYP1

Parts of Similar Triangles ALGEBRA Given , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT. Lesson 3 Ex2

Parts of Similar Triangles Since because they are alternate interior angles. By AA Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar polygons, Substitution Cross products Lesson 3 Ex2

Parts of Similar Triangles Distributive Property Subtract 8x and 30 from each side. Divide each side by 2. Now find RQ and QT. Answer: RQ = 8; QT = 20 Lesson 3 Ex2

A. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC. Lesson 3 CYP2

B. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find CE. Lesson 3 CYP2

What is the height of the Sears Tower? Indirect Measurement INDIRECT MEASUREMENT Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 P.M. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at that time. What is the height of the Sears Tower? Lesson 3 Ex3

Indirect Measurement Since the sun’s rays form similar triangles, the following proportion can be written. Now substitute the known values and let x be the height of the Sears Tower. Substitution Cross products Lesson 3 Ex3

Interactive Lab: Cartography and Similarity Indirect Measurement Simplify. Divide each side by 2. Answer: The Sears Tower is 1452 feet tall. Interactive Lab: Cartography and Similarity Lesson 3 Ex3

INDIRECT MEASUREMENT On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 feet 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot? A. 196 ft B. 39 ft C. 441 ft D. 89 ft Lesson 3 CYP3