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D. N. A. 1) Are the following triangles similar? A W 42 18 M 36 12 R 6 14 C T 2) Find the value of x and y. P 18 Q y PQRS~CDAB 12 24 D 16 S A 32 R C x B Similar Triangles Chapter 7-3 • Identify similar triangles. • Use similar triangles to solve problems. Standards 4.0 Students prove basic theorems involving congruence and similarity. (Key) Standard 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Triangle Similarity is: Lesson 3 TH2 Writing Proportionality Statements Given BTW ~ ETC • Write the Statement of Proportionality • Find mTEC E • Find TE and BE 20 EC TE TC BW TB TW mTEC = mTBW = 79o EC TE 3 x BW TB 12 20 T 3 34o C 79o TE 5 B 12 EB 20 5 15 W 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, K then JKL ~ XYZ. Y J L 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 Q A C P R Which of the following three triangles are similar? B 9 6 A H E 4 D 6 8 14 6 12 J 10 C F ABC and FDE? Longest Sides Shortest Sides Remaining Sides G ABC~ FDE SSS ~ Thm Scale Factor = 3:2 Which of the following three triangles are similar? B 9 6 A H E 4 D 6 8 14 6 12 J 10 C F ABC and GHJ Longest Sides G Shortest Sides ABC is not similar to DEF 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 Y J L X Z ass Pantograph Prove RTS ~ PSQ S S (reflexive prop.) S SP SQ SR ST 4 5 4(20) 5(16) 16 20 80 80 4 P 12 5 Q 15 SPQ SRT SAS ~ Thm. R T Are the two triangles similar? N P NQ 12 4 QT 9 3 PQ 15 3 RQ 10 2 15 12 Q 10 R NQP TQR 9 T Not Similar How far is it across the river? 2 yds x yards 5 yds 2 5 42 x 2x = 210 x = 105 yds 42 yds Determine the similar triangles. 5. Find x, AC, and A ED. x 1 E 15 B 12 x5 6. Find x, JL, and LM. J C x 18 16 D N L 4 x 3 M K 7. Find x, EH, and EF. E x5 H 9 F 6 12 G 9 D Are Triangles Similar? In the figure, , and ABC and DCB are right angles. Determine which triangles in the figure are similar. Are Triangles Similar? by the Alternate Interior Angles Theorem. Vertical angles are congruent, Answer: Therefore, by the AA Similarity Theorem, ΔABE ~ ΔCDE. 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 Parts of Similar Triangles ALGEBRA Given , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT. 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 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 A. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC. A. 2 B. 4 C. 12 D. 14 Lesson 3 CYP2 B. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find CE. A. 2 B. 4 C. 12 D. 14 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? 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 Indirect Measurement Simplify. Divide each side by 2. Answer: The Sears Tower is 1452 feet tall. Interactive Lab: Cartography and Similarity 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 Homework Chapter 7-3 • Pg 400 7 – 17, 21, 31 – 38