5 Minute Check on Lesson 6 2 Transparency

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 (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 m A = m P m B = m R

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 1 a 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 1 b 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 2 a ALGEBRA: Given RS // UT, RS=4, RQ=x+3, QT=2 x+10, UT=10, find RQ and QT Since because they are alternate interior angles. By AA Similarity, Using the definition of similar polygons,

Example 2 a cont Substitution Cross products Distributive Property Subtract 8 x and 30 from each side. Divide each side by 2. Now find RQ and QT. Answer:

Example 2 b ALGEBRA Given AB // DE, AB=38. 5, DE=11, AC=3 x+8, and CE=x+2, find AC and CE. Answer:

Example 3 a 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? Assuming that the sun’s rays form similar ∆s, the following proportion can be written.

Example 3 a 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 3 b 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 D C 5 W D Similar Triangles (determine if similar and list in proper order) E A P W x+3 6 V 35° C 85° B T 40° F R Z Q S N 6 x+1 L x-3 4 G A 10 y+1 8 B 10 1 S 11 x - 2 W M