Transcript Section 3
5-Minute Check on Lesson 6-2 Transparency 6-3 1. Determine whether the triangles are similar. Justify your answer. Yes: corresponding angles corresponding sides have same proportion 2. The quadrilaterals are similar. Write a similarity statement and find the scale factor of the larger to the smaller quadrilateral. ABCD ~ HGFE Scale factor = 2:3 3. The triangles are similar. Find x and y. x = 8.5, y = 9.5 4. Standardized Test Practice: Which one of the following statements is always true? A Two rectangles are similar B Two right triangles are similar C Two acute triangles are similar D Two isosceles right triangles are similar Click the mouse button or press the Space Bar to display the answers. Lesson 6-3 Similar Triangles Objectives • Identify similar triangles • Use similar triangles to solve problems Vocabulary • None new Theorems • Postulate 6.1: Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar • Theorem 6.1: Side-Side-Side (SSS) Similarity If all the measures of the corresponding sides of two triangles are proportional, then the triangles are similar Theorems Cont • Theorem 6.2: Side-Angle-Side (SAS) Similarity If the measures of two sides of a triangle are proportional to the measures of two corresponding side of another triangle and the included angles are congruent, then the triangles are similar • Theorem 6.3: Similarity of triangles is reflexive, symmetric, and transitive – Reflexive: ∆ABC ~ ∆ABC – Symmetric: If ∆ABC ~ ∆DEF, then ∆DEF ~ ∆ABC – Transitive: If ∆ABC ~ ∆DEF and ∆DEF ~ ∆GHI, then ∆ABC ~ ∆GHI AA Triangle Similarity A P Third angle must be congruent as well (∆ angle sum to 180°) Q From Similar Triangles B Corresponding Side Scale Equal C AC AB BC ---- = ---- = ---PQ PR RQ R If Corresponding Angles Of Two Triangles Are Congruent, Then The Triangles Are Similar mA = mP mB = mR SSS Triangle Similarity A P From Similar Triangles Corresponding Angles Congruent Q A P B R C Q B C R If All Three Corresponding Sides Of Two Triangles Have Equal Ratios, Then The Triangles Are Similar AC AB BC ---- = ---- = ---PQ PR RQ SAS Triangle Similarity A P Q B C R If The Two Corresponding Sides Of Two Triangles Have Equal Ratios And The Included Angles Of The Two Triangles Are Congruent, Then The Triangles Are Similar AC AB ---- = ---PQ PR and A P Example 1a In the figure, AB // DC, BE = 27, DE= 45, AE = 21, and CE = 35. Determine which triangles in the figure are similar. Since AB ‖ DC, then BAC DCE by the Alternate Interior Angles Theorem. Vertical angles are congruent, so BAE DEC. Answer: Therefore, by the AA Similarity Theorem, ∆ABE ∆CDE Example 1b In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5. Determine which triangles in the figure are similar. I Answer: Example 2a ALGEBRA: Given RS // UT, RS=4, RQ=x+3, QT=2x+10, UT=10, find RQ and QT Since because they are alternate interior angles. By AA Similarity, Using the definition of similar polygons, Example 2a cont Substitution Cross products Distributive Property Subtract 8x and 30 from each side. Divide each side by 2. Now find RQ and QT. Answer: Example 2b ALGEBRA Given AB // DE, AB=38.5, DE=11, AC=3x+8, and CE=x+2, find AC and CE. Answer: Example 3a INDIRECT MEASUREMENT Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12foot 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? Assuming that the sun’s rays form similar ∆s, the following proportion can be written. Example 3a cont Now substitute the known values and let x be the height of the Sears Tower. Substitution Cross products Simplify. Answer: The Sears Tower is 1452 feet tall. Example 3b 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? Answer: 196 ft Summary & Homework • Summary: – AA, SSS and SAS Similarity can all be used to prove triangles similar – Similarity of triangles is reflexive, symmetric, and transitive • Homework: – Day 1: pg 301-302: 6-8, 11-15 – Day 2: pg 301-305: 9, 18-21, 31 Ratios: QUIZ Prep 1) x 3 = 12 4 2) x - 12 x + 7 = -4 6 3) 28 7 = z 3 4) 14 x+2 = 10 5 Similar Polygons H K A 12 C J B 16 x+1 D C 5 W D Similar Triangles (determine if similar and list in proper order) P W E x+3 A 6 V 35° C 85° B T 40° N 6 L x-3 4 G A 10 y+1 8 B 10 S Z Q F R S 1 11x - 2 W M