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Module 8 Lesson 3 – Similar Triangles We are going to cover three parts in this section. By the end of this section you will be able to; 1. Identify Similar Triangles by AA Similarity, SSS Similarity, and SAS Similarity, 2. Use the Special Segments Theorems to solve problems. Similar Triangles There are three ways to identify if two triangles are similar. Angle-Angle (AA~) Similarity Postulate – If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example: Side-Side-Side (SSS~) Similarity Theorem – If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. Example: . Side-Angle-Side (SAS~) Similarity – If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. Example: BC AB and B E, so ABC ~ DEF . EF DE 4 1 6 1 ; and B E, so ABC ~ DEF 8 2 12 2 Example 1: A. B. C. Example A – demonstrates the SAS Similarity Postulate because the two sides of the first triangle are proportional to the two sides of the second triangle and the included angles are congruent. Example B – AA Similarity because the two angles of the the first triangle are congruent to the two angles of the second triangle. (If you look closely you notice the vertical angles in this image and remember that vertical angles are congruent. Example C – SSS Similarity because the corresponding sides of the two triangles are proportional. When looking at a set of triangles, to set up your own proportions, you place the smallest side over the smallest side and largest over largest, etc. Then test to see if the ratios are equivalent. = Now You Try For questions 1 – 6, if triangles can be proved similar, write the postulate or theorem that would be used to prove it and write the similarity statement for numbers 2 and 3. 1. 2. 3. T 4. 5. 6. Parts of Similar Triangles Special Segments of Similar Triangles Theorems Altitude If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. Example: Altitudes Examples: If the two triangles are similar, find x. A. B. Angle Bisector If two triangles are similar, then the measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides. Example: Angle Bisector Example: If the two triangles are similar, find x. Median If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. Example: Median Examples: If the two triangles are similar, find x. A. B. Now You Try: If the two triangles are similar, find x or MN. 1. 3. 2. Find MN.