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Name: ______________________________________________________________ Date: _______________________ Ch 6-4. Period: _______ Chapter 6: Similar Triangles Topic 4: Intro to Similar Triangle Proofs Definition of Similar Triangles β Two triangles are similar if and only if the corresponding sides are in _________________________ and the corresponding angles are _________________________. If: βπ΄π΅πΆ ~ βπ·πΈπΉ Μ Μ Μ Μ π΄π΅ Μ Μ Μ Μ π·πΈ Sides: Angles: = Μ Μ Μ Μ π΅πΆ Μ Μ Μ Μ πΈπΉ = Μ Μ Μ Μ π΄πΆ Μ Μ Μ Μ π·πΉ <π΄ β <π· <π΅ β <πΈ < πΆ β < πΉ Three Methods to prove Triangles are Similar β 1) __________________________ If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. If: <π΄ β <π· <π΅ β <πΈ Then: βπ΄π΅πΆ ~ βπ·πΈπΉ 2) __________________________ If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar. If: Μ Μ Μ Μ π΄π΅ Μ Μ Μ Μ π·πΈ = Μ Μ Μ Μ π΅πΆ Μ Μ Μ Μ πΈπΉ = Μ Μ Μ Μ π΄πΆ Μ Μ Μ Μ π·πΉ Then: βπ΄π΅πΆ ~ βπ·πΈπΉ 3) __________________________ If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangle are similar. If: Μ Μ Μ Μ π΄π΅ Μ Μ Μ Μ π·πΈ Μ Μ Μ Μ π΄πΆ = π·πΉ Μ Μ Μ Μ <π΄ β <π· Then: βπ΄π΅πΆ ~ βπ·πΈπΉ Ch 6-4. Practice: 1) How is βπ΄π΅πΈ ~ βπ·πΆπΈ? 2) Prove βπ΄π΅πΆ ~ βπππ using the given information. Show why or why not. 3) Determine if βπ΄π΅πΆ ~ βπ·πΈπΉ? 4) Prove βπΈπ΄π· ~ βπΆπ΄π΅ using the given information. Show why or why not. Ch 6-4. 5) Is βπΈπ·πΆ ~ βπ΄π΅πΆ? 6) Given: Μ Μ Μ Μ π΄π΅ = 10 < π΄ = 40 Μ Μ Μ Μ = 15 π΄πΆ Μ Μ Μ Μ π·πΈ = 20 < π· = 40 Μ Μ Μ Μ = 30 π·πΉ Prove: βπ΄π΅πΆ ~ βπ·πΈπΉ 7) If βπ ππ ~ βπ΄π΅πΆ, π < π΄ = π₯ 2 β 8π₯, π < πΆ = 4π₯ β 5, and π < π = 5π₯ + 30, find π < πΆ. _____ 8) In triangles ABC and DEF, AB = 4, AC = 5, DE = 8, DF = 10 and <A β <D. Which method could be used to prove βπ΄π΅πΆ ~ βπ·πΈπΉ? (1) AA (2) SAS (3) SSS (4) ASA Ch 6-4. _____ 9) Scalene triangle ABC is similar to triangle DEF. Which statement is false? (1) π΄π΅ βΆ π΅πΆ = π·πΈ βΆ πΈπΉ (2) π΄πΆ βΆ π·πΉ = π΅πΆ βΆ πΈπΉ (3) < π΄πΆπ΅ β < π·πΉπΈ (4) < π΄π΅πΆ β < πΈπ·πΉ _____ 10) In the diagram, βπ΄π΅πΆ ~ βπ ππ. Which statement is not true? π΄π΅ π΅πΆ (1) < π΄ β < π (2) π π = ππ (3) π΄π΅ π΅πΆ = ππ π π _____ 11) In βπ΄π΅πΆ and βπ·πΈπΉ, (1) π΄πΆ = π·πΉ (3) < π΄πΆπ΅ β < π·πΉπΈ (4) π΄πΆ π·πΉ = πΆπ΅ . πΉπΈ π΄π΅+π΅πΆ+π΄πΆ π π+ππ+π π = π΄π΅ π π Which additional information would prove βπ΄π΅πΆ ~ βπ·πΈπΉ? (2) πΆπ΅ = πΉπΈ (4) < π΅π΄πΆ β < πΈπ·πΉ Ch 6-4. Topic 4 Homework: Intro To Similar Triangle Proofs 1) Find the scale factor that proves βπ΄π΅πΆ ~ βπ·πΈπΉ. Then show that this scale factor holds true for every side of the dilated figure. 2) How is βπ΄π΅πΆ ~ βπΊπ»πΌ? 3) Determine if βπ΄π΅πΆ ~ βπ·πΈπΉ or not. Show a proof of why or why not. Ch 6-4. 4) Determine if βπ΄π΅πΆ ~ βπ·πΈπΉ? 5) Is ~ π₯π·πΈπΉ? Prove why or why not. Μ Μ Μ Μ Μ = 15 6. Given: ππ π < π = 70° Μ Μ Μ Μ ππ = 21 Μ Μ Μ Μ ππ = 5 π < π = 70° Μ Μ Μ Μ ππ = 7 Prove: π₯πππ ~ π₯πππ Ch 6-4. 7) In the diagram, βπ΄π΅πΆ ~ βπ·πΈπΉ, DE = 4, AB = x, AC = x + 2, and DF = x + 6. Determine the length of AC. Review Section: _____8) βπ΄π΅πΆ is shown in the diagram. If DE joins the midpoints of ADC and AEB, which statement is not true? 1 (1) DE = 2 πΆπ΅ (2) DE // CB (3) π΄π· π·πΆ = π·πΈ πΆπ΅ (4) βπ΄π΅πΆ ~ βπ΄πΈπ· _____9) Given that ABCD is a parallelogram, a student wrote the proof on the right to show that a pair of opposite angles are congruent. What is the reason justifying that < π΅ β < π·? (1) Opposite angles in a quadrilateral are congruent (2) Parallel lines have congruent corresponding angles (3) Corresponding parts of congruent triangles are congruent (4) Alternate interior angles in congruent triangles are congruent _____10) In βπ΄π΅πΆ, DE // BC. If AB = 10, AD = 8, and AE = 12, What is the length of EC? (1) 6 (2) 2 (3) 3 (4) 15 _____11) What is the slope of a line perpendicular to the line whose equation is 20π₯ β 2π¦ = 6? 1 (1) β10 (2) β 10 (3) 10 (4) 1 10 Ch 6-4. Not β So β Formal Similar Triangle Proofs HW 1) Scale factor = 2 3) βπ΄π΅πΆ is not ~ βπ·πΈπΉ πππ ~ 2) πππ~ 4) π΄π΄~ 5) πππ~ 7) π₯ = 2 6) ππ΄π ~ π΄πΆ = 4 Review Section: 8) (3) 10) (3) 9) (3) 11) (2)