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**By: THE “A” SQUAD (Annie and Andrew)**

CHAPTER 8: Sections By: THE “A” SQUAD (Annie and Andrew)

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Vocabulary Terms Dilation: A transformation that is not rigid and preserves the shape of an object despite size changes. Scale Factor: The number that both X and Y are multiplied by to get the Image. Similar Figures: Figures are similar when the image of the other is congruent by a dilation.

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**Postulates and Theorems**

Polygon Similarity Postulate: Two polygons are similar if and only if there is a way of setting up a correspondence between their sides and angles such that each pair of corresponding angles is congruent and each pair of corresponding sides is proportional. Angle-Angle Postulate: If two angles are congruent to two angles of another triangle, then the triangles are similar. Side-Side-Side Theorem: If three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. Side-Angle-Side Theorem: If two sides and their included angle are proportional/congruent respectively, then the triangles are similar.

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**Examples: A D 14 Given: AB= 8; BC= 3; AC= 14; DE= 4; EF= 1.5; DF= 7**

Prove: Triangle ABC ~ Triangle DEF 8 7 4 E F B C 3 1.5 AB/DE= 8/4= 2 BC/EF= 3/1.5= AC/DF= 14/7= 2 Thus, the sides of the triangles are proportion and, by the SSS Similarity Theorem, Triangle ABC ~ Triangle DEF.

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More Examples: Given: <x = 25; <g = 25; xy = 2; xz = 3.6; fg= 3; gh = 5.4 Prove: Triangle XYZ ~ Triangle GFH g x 25 25 2 3.6 5.4 3 y z f h YX/FG = 2/3 XZ/GH = 3.6/5.4 = 2/3 Thus, the sides of the triangle are proportional, and the included angles of these sides are congruent. By the SAS Similarity Theorem, Triangle QRS ~ Triangle UTV.

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Daily Application

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**Quiz: Find the image of (3,3) for a dilation with scale factor 1/3.**

Write a similarity statement for the two triangles. (1,1) E ABC ~ FED ACB ~ FDE BAC ~ EFD BCA ~ EDF CBA ~ DEF CAB ~ DFE B 54 65 D F 61 A C

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**Quiz Continued 3. Find x and y for these similar rectangles.**

18 6 x y 4 4 6 18 6/18 = 4/x 6x = 72 x =12 y = 12

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7.4 A Postulate for Similar Triangles. We can prove that 2 triangles are similar by showing that all 3 corresponding angles are congruent, and all 3 sides.

7.4 A Postulate for Similar Triangles. We can prove that 2 triangles are similar by showing that all 3 corresponding angles are congruent, and all 3 sides.

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