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Transcript
Chapter 4 Congruent Triangles 4-1 Objectives a) _____________________________________________________________ _____________________________________________________________ b) _____________________________________________________________ _____________________________________________________________ Congruent or not congruent? Congruent or not congruent? What are congruent polygons? • Come up with a definition of congruent polygons based on the previous discussion. How do we name congruent polygons? • What would our congruence statement look like? • It should be very clear from the statement which parts of the two triangles are corresponding. Find the missing angle measures • ∆ABC ∆DEF • If mA = 34, what is mD? • If BC = 7, what is EF? • If mA = 30 and mB = 50, what is mF? Explain how you know. How could we be sure these are congruent? • Third Angles Theorem Proving the rd 3 Angles Theorem • Given: 𝐴 𝐷, 𝐵 𝐸 • Prove: 𝐶 𝐹 Proving Triangles Congruent Practice: P. 223-224 #32, 34, 39, 45 Lazy Lawrence • Lawrence works for a company that makes roof trusses, triangular pieces that support simple roofs. His job is to ensure that each and every roof truss that is made at the company is exactly the same size as all the others. Because Lawrence is so lazy and likes to avoid as much work as possible, he wants to find the easiest possible way to do his job. Help Lawrence find the easiest way to show that two triangular trusses are equal in measure with as little work as possible. What is the minimum amount of parts that would need to be congruent to ensure two triangles are congruent? (work with a partner) • 1 side (S)? True or False? • If two triangles have one side of equal measure, then the triangles are congruent. • 1 angle (A)? True or False? • If two triangles have one angle of equal measure, then the triangles are congruent. What is the minimum amount of parts that would need to be congruent to ensure two triangles are congruent? (work with a partner) • 2 sides (SS)? True or False? • If two triangles have two sides of equal measure, then the triangles are congruent. • 2 angles (AA)? True or False? • If two triangles have two angles of equal measure, then the triangles are congruent. What is the minimum amount of parts that would need to be congruent to ensure two triangles are congruent? (work with a partner) • 1 side and 1 angle (SA)? True or False? • If two triangles have one side and one angle of equal measure, then the triangles are congruent. • Find and test the remaining combinations (involving 3 parts of the triangle) to determine which hypotheses would be true. Record on the next page • Things to consider: Is SAS different than SSA? Is AAS different than ASA? Record findings here Which hypotheses were true? What about Angle-Angle-Side (AAS)? • Proof? What about right triangles? • Proof? 4-2 Objectives a) _____________________________________________________________ _____________________________________________________________ 4-3 Objectives a) _____________________________________________________________ _____________________________________________________________ 4-6 Objectives a) _____________________________________________________________ _____________________________________________________________ Practice: p. 231 #16, 17, 29, 31 p. 239 #15, 19, 20 4-4 Objectives a) _____________________________________________________________ _____________________________________________________________ How can you prove two parts of a triangle are congruent? Complete proofs using CPCTC • Given: 𝑌𝐴 ≅ 𝐵𝐴 B Y • Prove: 𝐴𝑍 ≅ 𝐴𝐶 Practice: p. 247 #9, 18, 19 4-5 Objectives a) _____________________________________________________________ _____________________________________________________________ List what you know about: • Isosceles Triangles • Equilateral Triangles Can you prove the isosceles triangle theorem? • If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. • What would the converse of this theorem be? Corollaries – proved easily using another theorem • Corollary to Isosceles Triangle Theorem (Equilateral Triangles) • Corollary to Converse of the Isosceles Triangle Theorem (Equiangular Triangles) Practice: pp. 254-255 #10-12, 23, 25, 30-32 4-7 Objectives a) _____________________________________________________________ _____________________________________________________________ b) _____________________________________________________________ _____________________________________________________________ Can we prove these overlapping triangles congruent? Given: RE TC , REP TCG, PC GE Prove: REP TCG R T P C G E Can you prove these triangles congruent? Practice: pp. 269-270 #15-18, 21, 22