6.2 Similar Triangles or Not? 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 35 100 = 𝑥 230 𝑥= 35∙230 100 𝑥=80.5 𝑙𝑏𝑠. 𝑜𝑓 𝑠𝑢𝑔𝑎𝑟

6.2 Similar Triangle Theorems

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) Collaborate #1 (2 Minutes) ∆𝑅𝑆𝑇~∆𝑊𝑋𝑌 Corresponding Angles ∠𝑅≅∠𝑊,∠𝑆≅∠𝑋,∠𝑇≅∠𝑌 Corresponding Proportional Sides 𝑅𝑆 𝑊𝑋 = 𝑆𝑇 𝑋𝑌 = 𝑅𝑇 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 𝑋𝑌 = 33 22 = 3 2 𝑉𝑊 𝑌𝑍 = 24 16 = 3 2 𝑈𝑊 𝑋𝑍 = 36 24 = 3 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

AA~

Ratio of sides = 1 3 SAS~

Formative Assessment AA~ SSS~ SAS~ Skills Practice 6.2 Problem Set: Pg. 519-526 (1-20) AA~ SSS~ SAS~