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Lesson 3 COMMON CORE MATHEMATICS CURRICULUM β Base Angles of Isosceles Triangles U3 GEOMETRY Name__________________________________________ Period______ Date____________ Lesson 3: Base Angles of Isosceles Triangles Classwork Opening Exercise Describe the additional piece of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria. 1. Given: π΄π΅ = π·πΆ __________________________, 2. Prove: β³ π΄π΅πΆ β β³ π·πΆπ΅. Given: AB = RT π΄π΅ β₯ π π __________________________. Prove: β³ π΄π΅πΆ β β³ π ππ. 1 COMMON CORE MATHEMATICS CURRICULUM β Base Angles of Isosceles Triangles Lesson 3 U3 GEOMETRY Discussion Today we examine a geometry fact that we already accept to be true. We are going to prove this known fact in two ways: (1) by using transformations, and (2) by using SAS triangle congruence criteria. Here is isosceles triangle β³ π΄π΅πΆ. We accept that an isosceles triangle, which has (at least) two congruent sides, also has congruent base angles. Label the congruent angles in the figure. Now we will prove that the base angles of an isosceles triangle are always congruent. Prove Base Angles of an Isosceles are Congruent: Transformations Given: Isosceles β³ π΄π΅πΆ, with π΄π΅ = π΄πΆ. Prove: β π΅ β β πΆ. βββββ of β π΄, where π· is the intersection of Construction: Draw the angle bisector π΄π· the bisector and π΅πΆ. We need to show that rigid motions will map point π΅ to point πΆ and point πΆ to point π΅. Let Ξ be the reflection through πΏπ΄π· . Through the reflection, we want to demonstrate two pieces of βββββ maps to π΄πΆ βββββ and (2) π΄π΅ = π΄πΆ. information that map π΅ to point πΆ and vice versa: (1) π΄π΅ Since A is on the line of reflection, β‘ββββ AD, Ξ(A) = A. Reflections preserve angle measures, |Ξβ BAD| = βββββ ) = βββββ |β CAD| and so Ξ(AB AC. Reflections also preserve lengths of segments, therefore the reflection of AB will still have the same length as AB, or, |Ξ(AB)| = |AB|. By hypothesis AB = AC, so the length of the reflection will also be equal to AC, or |Ξ(AB)| = |AC|. Then Ξ(B) = C. Using similar reasoning, we can show that Ξ(C) = B. βββββ ) = CA βββββ and Ξ(BC βββββ ) = CB βββββ . Again, since reflections preserve angle Reflections map rays to rays, so Ξ(BA measures, Ξ(β ABC) = β ACB. We conclude that β π΄π΅πΆ = β π΄πΆπ΅ (β π΅ β β πΆ). 2 COMMON CORE MATHEMATICS CURRICULUM β Base Angles of Isosceles Triangles Lesson 3 U3 GEOMETRY Prove Base Angles of an Isosceles are Congruent: SAS Given: Isosceles β³ π΄π΅πΆ, with π΄π΅ = π΄πΆ. Prove: β π΅ β β πΆ. Construction: Draw the angle bisector βββββ π΄π· of β π΄, where π· is the intersection of the bisector and π΅πΆ. We are going to use this auxiliary line towards our SAS criteria. Exercises 1. Given: π½πΎ = π½πΏ. πΎπ = π πΏ. Prove: π½π β₯ πΎπΏ. 2. Given: π΄π΅ = πΆπ΅, ππ΅ = ππΆ. Prove: π΄π bisects β π΅π΄πΆ. 3 COMMON CORE MATHEMATICS CURRICULUM β Base Angles of Isosceles Triangles Lesson 3 U3 GEOMETRY 3. Given: π½π = π½π, πΎπ = πΏπ. Prove: β³ π½πΎπΏ is isosceles. 4. Given: β³ π΄π΅πΆ, with β πΆπ΅π΄ = β π΅πΆπ΄. Prove: π΅π΄ = πΆπ΄. (Converse of Base Angles of Isosceles Triangle, Abbreviation: base β s converse) Hint: Use a transformation after constructing an auxiliary line (which type of construction makes sense? _________________________). 4 COMMON CORE MATHEMATICS CURRICULUM β Base Angles of Isosceles Triangles Lesson 3 U3 GEOMETRY 5. Given: β³ π΄π΅πΆ, with ππ is the angle bisector of β π΅ππ΄, and π΅πΆ β₯ ππ. Prove: ππ΅ = ππΆ. Hint: β πΏππ¨ = β π©πͺπ corr. β π, π©πͺ β₯ πΏπ 5