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Name: ______________________________ Geometry Per ____ Lesson 9-6 Notes Date: _________________ 8-9 Proving Similar Triangles Day 2 Learning Goals: How can we prove two triangles are similar? How can we use similarity to prove proportions? How can we use similarity to prove cross product equivalency? Today we are going to continue with similarity proofs. From yesterday, we learned how to prove triangles are similar. Now, we are going to extend that and prove proportions and equal cross products. We use 3 tiers to guide us through proofs of and/or involving similarity: *We can say that these two triangles are similar by the ____________ shortcut. *Since they are similar we can set up the following proportion: π΄π΅ π΄πΆ = ππ *Take it one step further! What cross products do we get when we cross multiply? Letβs try one: Given is an isosceles triangle with base AC and BD is perpendicular to AC, prove AD β BC = BA β CD Where should we start? What tier do we need to use? What triangle are we going to prove to be similar? Whatβs our plan? Justify at all steps! *What is given to us? *What can we infer from the givens? Statement 1. is isosceles triangle with base AC and BD is perpendicular to AC. Reason 1. π. β BDA and β BDC are right angles. 2. π. β BDA β β BDC 3. All right angles are congruent. π. β A β β π©πͺπ¨ π. 5. βπ©π«π¨ and βπ©π«πͺ are ________________. 5. Perfect Practice Makes Perfect! You are going to make a fruit salad! Your salad needs to consist of at least two apples, 2 pineapples, and 2 watermelons! You get to pick which oneβs you want to do! Apples - you need at least 2! 1. Given: AB//DE Prove: βπ·πΆπΈ ~ βπ΅πΆπ΄ Statement Reason Pre-proof work Do I need to re-draw my triangles separately? Did I mark my diagram? Do I have a plan? 2. Given: βABC with BA β BC and BD is an angle bisector of<B. Prove: βABD ~ βCBD *Careful! What does it mean for BD to be an angle bisector? 3. Given: βABC is isosceles with base AC; BD and EC are altitudes. Prove: βπ΅π·πΆ ~ βπΆπΈπ΄. (*Hint! What do you remember about the altitude of an isosceles triangle? Does it have any other special properties?) Statements Reasons 4. Givens: βFGH is an isosceles triangle with legs FG and HG. GK bisects β FGH. Prove: βFGK ~ HGK PINEAPPLES - you need at least 2! 1. Given: WA // CH and WH and AC intersect at point T. πΎπ» πΎπ¨ Prove: = π―π» π―πͺ 2. Given: HW//TA and HY//AX, prove π¨πΏ π―π = π¨π» π―πΎ hH lines! Hint: extend Pre-Proof work: 3. Given: Trapezoid ABCD with bases BC and AD π΅πΆ Prove: π΄π· = πΈπΆ πΈπ΄ a) What tier type of question is this? b) Based on the given information mark your diagram appropriately (In a trapezoid the bases are _______________) c) Determine the 2 triangles we are looking to prove similar based on the sides we are working with (redraw the triangles below). WATERMELON - you need at least 2! 1. Given AC β₯ BD and DE β₯ AB. Prove: AC * BD = DE * AB Where should we start? What tier do we need to use? What triangle are we going to prove to be similar? Redraw Triangles! Whatβs our plan? Justify at all steps! *What is given to us? *What can we infer from the givens? Statement 1. AC β₯ BD and DE β₯ AB. Reason 1. 2. π. β ________and β ________ are right angles. π. β ______ β β ______ 3. All right angles are congruent. 4. 4. π. β and β 6. = are ________________. 7. AC * BD = DE * AB 2. Μ Μ Μ Μ β₯ π΄π΅ Μ Μ Μ Μ Μ and πΆπ΅ Μ Μ Μ Μ β₯ πΆπ΄ Μ Μ Μ Μ Μ Given: πΆπ· 5. π. 7. Prove: BC * DB = AC * CD Pre-proof work Do I need to re-draw my triangles separately? Did I mark my diagram? Do I have a plan? (What tier should I use?) 3. Given CB β₯ BA, CD β₯ DE, prove AB * CD = DE * CB