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Warmup Which is the correct definition of isosceles? a.) a triangle with at least 2 congruent sides b.) a triangle with exactly 2 congruent sides 4.5 Isosceles Triangles LEQ: How can we use properties of isosceles triangles to construct proofs? Word Bank Leg Leg Vertex Base Base Angles Given Proof: ABC is isosceles. Prove m<A=m<B. Statements 1.) AC=BC C Reasons 1.) Definition of Isosceles *Construct CD so that it bisects <C* 2.)m<ACD=m<BCD 2.)Defn. of bisect 3.) CD=CD 3.) reflexive property 4.) ACD BCD 4.) SAS What can you conclude about <A and <B? 5.) <A <B 5.) CPCTC A D B Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, then the base angles are congruent as well. Converse: If the base angles of a triangle are congruent, then the 2 opposite legs of the triangle are congruent as well. What kind of statement is this? Given BC=AC, answer the following: a.) If m<1=140, m<2=_____, C m<3=______ and m<4=______ 1 2 b.) If m<4=65, m<2=_____, m<3=______and m<1=______. A 3 4 B Corollaries (small theorem that follow a big theorem) • An equilateral triangle is also equiangular. • An equiangular triangle has all 60 angles. Perpendicular bisector: • Draw an isosceles triangle • Bisect the vertex • What do you notice about the intersection of the bisector and the base? Corollary 3 The bisector of the vertex of an isosceles triangle is a perpendicular bisector of the base. (divides the base in half and is perpendicular to the base) Try examples 1-3. Ex. 1 An exterior angle of an isosceles triangle has the measure 11o degrees. What are the 2 possible sets of measures for the angles of the triangle?40,70,70 and 70,55,55 110 40 110 70 70+x+x=180 x=55 x x 70 70 110 Ex. 2: Find the values of m and n: m+n= 45 60 m= n m