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PROVING TRIANGLE SIMILARITY Unit 2 lesson 2 OBJECTIVES • determine whether two triangles are similar • prove or disprove triangle similarity using similarity shortcuts (AA, SSS, SAS) SIMILAR TRIANGLES Triangles are SIMILAR if their corresponding sides are proportional and corresponding angles are congruent. (same shape, different size) Similarity Statement: ∆𝑷𝑷𝑷𝑷𝑷𝑷~∆𝑫𝑫𝑫𝑫𝑫𝑫 SIMILAR TRIANGLES Congruent triangles have corresponding parts with both angle measures and side lengths that are the same. 1. a) If 2 triangles are congruent, are they similar? Explain. b) If 2 triangles are similar, are they congruent? Explain. PROOFS • Paragraph proofs are statements written out in complete sentences in a logical order to show an argument • Flow proofs are graphical methods of presenting logical steps used to show an argument • Two-Column Proofs are numbered statements and corresponding reasons that show the argument in a logical order. (This is the type of proof we will be working with) EXAMPLE (TWO COLUMN PROOF) Given: The figure to the right is a parallelogram, Prove: 𝑚𝑚∠𝐷𝐷 = 𝑚𝑚𝑚𝑚𝑚. statements 1. is a parallelogram 2. 𝑚𝑚∠𝐷𝐷 = 𝑚𝑚𝑚𝑚𝑚 reasons 1. Given 2. If a quadrilateral is a parallelogram, then opposite sides are congruent SIMILAR TRIANGLES What is the minimum amount of information needed to prove that two triangles are similar? In other words, how much informations can I take away before we no longer know they are similar? ANGLE-ANGLE (AA) SIMILARITY SHORTCUT If two angles of one triangle are congruent (equal) to two angles of another triangle, then the triangles are similar. AA SIMILARITY Using a two column proof, prove that ∆𝐴𝐴𝐴𝐴𝐴𝐴~∆𝑋𝑋𝑋𝑋𝑋𝑋 STATEMENT REASON SIDE-SIDE-SIDE SIMILARITY SHORTCUT If the measure of the corresponding sides of two triangles are proportional (Scale Factor), then the triangles are similar. SSS SIMILARITY Using a two column proof, prove that the triangles are similar STATEMENT REASON SIDE-ANGLE-SIDE SIMILARITY SHORTCUT If the measures of two sides of a triangle are proportional (scale factor) to the measures of two corresponding sides of another triangle, and the included angles (between these corresponding sides) are congruent, then the triangles are similar. SAS SIMILARITY Prove that the triangles are similar. STATEMENT REASON EXAMPLE PROBLEM 1 Determine whether the triangles are similar. Prove or Disprove. STATEMENT REASON EXAMPLE PROBLEM 2 Determine whether the triangles are similar. Prove or disprove. STATEMENT REASON OBJECTIVES • determine whether two triangles are similar • prove or disprove triangle similarity using similarity shortcuts (AA, SSS, SAS) IN CLASS ASSIGNMENT & HOMEWORK