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Transcript
Section 8.3 Proving Triangles Similar By: Asad Ashraf What is Similarity • Similar figures are figures in which the shape is exactly the same, but the size is not. • A Dilation is an enlargement of a figure. It is still similar, however. • A Reduction is a reduction of a figure. This is also similar. • **Remember that similar figures are not necessarily congruent** Proving Triangles Similar • There are several methods to prove triangles similar. • These are very similar to the methods of proving triangles congruent. • The methods are AAA~, AA~, SAS~, and SSS~. AAA~ Postulate • AAA~ Postulate - If three angles of one triangle are congruent to three angles of a 2nd triangle then the triangles are similar. AA~ Theorem • AA~ if 2 angles of one triangle are congruent to two angles of a 2nd triangle then the triangles are similar. • This is proved by the No-Choice Theorem and AAA~ Theorem combined. SSS~ Theorem • SSS~ If 3 sides of one triangle are proportional to three sides of a 2nd triangle then the two triangles are similar. SAS~ Theorem • SAS ~ If two sides of one triangle are proportional to the corresponding two sides of a 2nd triangle and their included angles are congruent, then the triangles are similar. Sample Problems • Click Here • SSS~ and SAS~ Problems Sample Problems Are these triangles ~? Answer: Yes they are because Included angles were given congruent and the ratios of the sides are congruent as well. Practice Problems • Prove that an acute angle of one right angle is congruent to the vertex angle of an isosceles triangle, they are similar. • Always, Sometimes, Never – If 2 triangles are similar, they are congruent ____ – If 2 triangles are congruent, they are similar ____ – 2 rectangles are similar if neither is a square ____ – 2 right triangles are similar____ Answers: A, S, S, S Practice Problems • Are any 2 isosceles triangles similar? Show all work. Practice Problems • Are these triangles similar? Why or why not? Answer: NO Practice Problems • Given: XS and RY are Altitudes of RTS • Prove: Triangle TSX is similar to Triangle TYR Works Cited • Our Book • “Similar Triangles”. Mathwarehouse.com. May 27,2008.<http://www.mathwarehouse.com/geometry/similar /triangles/index.html>. • “Chapter 8”. Teacherweb.org. 2003. May 27,2008.<http://teacherweb.ftl.pinecrest.edu/wingjoa/My%2 0Webs/Geometry/chpt8.htm#Test%20Review%20%20%20Chapter%208.1-4>.