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GEOMETRY NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M2 Name:______________________________ Date:___________ Lesson 14: Similarity in Triangles Important notation: If 𝐴 and 𝐵 are similar, we write 𝐴~𝐵 where “~” denotes similarity. Example 1 Using the reflexive property we say 𝐴~𝐴. We said that for a figure 𝑨 in the plane, it must be true that 𝑨~𝑨. Why is this true? There are several different transformations that will map 𝑨 onto itself such as: A rotation of 0° or a rotation of 360°. A reflection of 𝐴 across a line and a reflection right back will achieve the same result. l A translation with a vector of length 0 also maps 𝐴 to 𝐴. A dilation with scale factor 1 will map 𝐴 to 𝐴, and any combination of these transformations will also map 𝐴 to 𝐴. Lesson 14: Date: Similarity 3/31/16 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 217 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Example 2 Using the symmetric property we say if 𝐴~𝐵, then 𝐵~𝐴. We say that for figures 𝑨 & 𝑩 in the plane so that 𝑨~𝑩, it must be true that 𝑩~𝑨. Why must this be true? This condition must be true because for any composition of transformations that maps 𝐴 to 𝐵, there will be a composition of transformations that can undo the first composition. For example, if: A translation by vector ⃑⃑⃑⃑⃑ 𝑋𝑌 maps 𝐴 to 𝐵, then the vector ⃑⃑⃑⃑⃑ 𝑌𝑋 will undo the transformation and map 𝐵 to 𝐴. A rotation of 90° in the counterclockwise direction can be undone by a rotation of 90° in the clockwise direction. A dilation by a scale factor of 𝑟 = 2 can be undone with a dilation by a scale factor of 1 𝑟 = 2. A reflection across a line can be undone by a reflection back across the same line. Example 3 Using the transitive property we can say if 𝐴~𝐵 and 𝐵~𝐶, then 𝐴~𝐶. Lesson 14: Date: Similarity 3/31/16 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 218 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Example 4 The following triangles are similar. a. similarity statement b. Given the similarity statement, find the scale factor for the ratio of the corresponding sides. congruent angles ABC ~ JLK c. corresponding sides AB BC AC JL LK JK A J B L C K Exercise 1 and 2 For each pair of similar triangles: a. c. 1. Write the similarity statement. b. Name the congruent angles. Write the correspondence between the sides. d. Determine the scale factor. a. similarity statement b. congruent angles c. corresponding sides 2. Given similar triangles ABC and DEF, complete a-d and find the missing side measures. 30 6 18 a. similarity statement Lesson 14: Date: b. congruent angles c. corresponding sides Similarity 3/31/16 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 219 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Name Date _____ Geometry M2L14 Similarity HW Period ____________ 1. Given similar triangles JKL and RST, a. Write the similarity statement b. Name the correspondence between all sides and all angles. c. Find the measure of the missing angles. 2. Given similar triangles ABC and DEF, a. Write the similarity statement 10 b. Name the correspondence between all sides and all angles. c. Find the scale factor. d. Find the measure of the missing sides. Lesson 14: Date: 3 6 Similarity 3/31/16 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 220 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.