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GEOMETRY 6.2 Similar Triangles or Not? 6.2 Similar Triangle Theorems • Objectives • Use constructions to explore similar triangle theorems • Explore the Angle-Angle (AA~) Similarity Theorem • Explore the Side-Side-Side (SSS~) Similarity Theorem • Explore the Side-Angle-Side (SAS~) Similarity Theorem • Key Terms • Included Angle and Included Side What do we need to set up to solve this problem? A PROPORTION Equal ratios that compare units 𝑙𝑏𝑠. 𝑝𝑒𝑜𝑝𝑙𝑒 35 𝑥 = 100 230 35 ∙ 230 𝑥= 100 𝑥 = 80.5 𝑙𝑏𝑠. 𝑜𝑓 𝑠𝑢𝑔𝑎𝑟 6.2 Similar Triangle Theorems Skipping 6.1 Problem 1: Using Two Angles • What does it mean for figures to be similar? • Two polygons are similar if all corresponding angles are congruent and the ratios of the measures of all corresponding sides are equal. (Pg. 446) • Pg. 452 Collaborate #1 (2 Minutes) ∆𝑅𝑆𝑇~∆𝑊𝑋𝑌 • Corresponding Angles • ∠𝑅 ≅ ∠𝑊, ∠𝑆 ≅ ∠𝑋, ∠𝑇 ≅ ∠𝑌 • Corresponding Proportional Sides • 𝑅𝑆 𝑊𝑋 𝑆𝑇 𝑅𝑇 = 𝑋𝑌 = 𝑊𝑌 Problem 1: Using Two Angles • We only need two congruent corresponding angles to say that two triangles are similar. • We can prove this by using constructions but we are not going to do those today • Skip 2-4 Construction Problem 1: Using Two Angles • Angle-Angle Similarity Theorem (AA~) • If two angles of one triangle are congruent to two angles of another triangles, then the triangles are similar. • Together #5 Problem 1: Using Two Angles • Collaborate 6-7 (2 Minutes) #6 No, the angles in the individual triangles are congruent, but we don’t know anything about transferring between the separate triangles. Problem 1: Using Two Angles #7 Yes, the angles in the individual triangles are congruent and the vertex angles are congruent which allows us to know that both sets of base angles are congruent. Problem 2: Using Two and Three Proportional Sides • Skip 1-6 Construction • Side-Side-Side Similarity Theorem (SSS~) • If the corresponding sides of two triangles are proportional, then the triangles are similar. Problem 2: Using Two and Three Proportional Sides • Collaborate 7-8 (3 Minutes) #7 If the lengths of the sides are known, then the measures of the angles in the triangle are known. (We could find them all: Later Lessons) Problem 2: Using Two and Three Proportional Sides 𝑈𝑉 33 3 = = 𝑋𝑌 22 2 𝑉𝑊 24 3 = = 𝑌𝑍 16 2 𝑈𝑊 36 3 = = 𝑋𝑍 24 2 The triangles are similar because the ratios of the corresponding sides are equal Problem 3: Using Two Proportional Sides and an Angle • Included Angle • An angle formed by two consecutive sides of a figure. • The angle between the two sides used. • Included Side • A line segment between two consecutive angles of a figure. • The side between the two angles used. • Skip 1-4 Construction Problem 3: Using Two Proportional Sides and an Angle • Side-Angle-Side Similarity Theorem (SAS~) • If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. Talk the Talk Pg. 460 Angle-Angle Side-Side-Side Side-Angle-Side Reminder: Parallel Lines Reminder: Reflexive Property When 2 lines are ∥, angles are either congruent or supplementary. Congruent to itself AA~ Ratio of 2 sets of sides = SAS~ 1 3 Formative Assessment • Skills Practice 6.2 • Problem Set: Pg. 519-526 (1-20) AA~ SSS~ SAS~