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GEOMETRY MODULE 1 LESSON 23 BASE ANGLES OF ISOSCELES TRIANGLES OPENING EXERCISE Describe the additional piece of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria. ο· Given: π΄π΅ = π·πΆ Additional Info Necessary: We need information to show congruent angles. Prove: βπ΄π΅πΆ β βπ·πΆπ΅ ο· Μ Μ Μ Μ β₯ π π Μ Μ Μ Μ Given: π΄π΅ = π π and π΄π΅ Additional Info Necessary: We need information to show congruent bases. Prove: βπ΄π΅πΆ β βπ ππ DISCUSSION Consider the isosceles triangle. We accept that an isosceles triangle, which has (at least) two congruent sides, also has congruent base angles. We can prove that πβ π΅ = πβ πΆ in two ways: Transformations and SAS. What transformation can we use to map β π΅ onto β πΆ? Reflection on the line of symmetry. MOD1 L23 1 Letβs prove β π΅ β β πΆ through SAS. To help with our proof, we construct an angle bisector. STEP JUSTIFICATION π΄π΅ = π΄πΆ Given π΄π· = π΄π· Reflexive Property πβ π΄π΅π· = πβ π΄πΆπ· Definition of Angle Bisector βπ΄π΅π· β βπ΄πΆπ· SAS β π΅ β β πΆ Corresponding angles of congruent triangles are congruent. MOD1 L23 2 PRACTICE 1. Given: π½πΎ = π½πΏ and Μ Μ Μ π½π bisects Μ Μ Μ Μ πΎπΏ. Μ Μ Μ β₯ πΎπΏ Μ Μ Μ Μ Prove: π½π STEP JUSTIFICATION Μ Μ Μ bisects πΎπΏ Μ Μ Μ Μ . π½πΎ = π½πΏ and π½π Given β πΎ = β πΏ Base angles of an isosceles triangle are congruent. πΎπ = πΎπΏ Definition of segment bisector βπ½π πΎ β βπ½π πΏ SAS β π½π πΎ β β π½π πΏ Corresponding angles of congruent triangles are congruent. MOD1 L23 πβ π½π πΎ + πβ π½π πΏ = 180° Linear pairs are supplementary. 2(πβ π½π πΎ) = 180° Substitution πβ π½π πΎ = 90° Division Μ Μ Μ π½π β₯ Μ Μ Μ Μ πΎπΏ Definition of perpendicular lines 3 2. Given: π΄π΅ = π΄πΆ, ππ΅ = ππΆ. Μ Μ Μ Μ bisects β π΅π΄πΆ Prove: π΄π 1 2 STEP JUSTIFICATION π΄π΅ = π΄πΆ Given πβ π΄π΅πΆ = πβ π΄πΆπ΅ Base angles of an isosceles triangle are equal in measure. ππ΅ = ππΆ 3 4 πβ ππ΅πΆ = πβ ππΆπ΅ Given Base angles of an isosceles triangle are equal in measure. 5 πβ π΄π΅πΆ = πβ π΄π΅π + πβ ππ΅πΆ Angle Addition πβ π΄πΆπ΅ = πβ π΄πΆπ + πβ ππΆπ΅ 6 πβ π΄π΅π = πβ π΄π΅πΆ β πβ ππ΅πΆ Subtraction πβ π΄πΆπ = πβ π΄πΆπ΅ β πβ ππΆπ΅ 7 πβ π΄π΅π = πβ π΄π΅πΆ β πβ ππ΅πΆ Substitution from lines 2 and 4 πβ π΄πΆπ = πβ π΄π΅πΆ β πβ ππ΅πΆ 8 πβ π΄π΅π = πβ π΄πΆπ Transitive 9 βπ΄π΅π β βπ΄πΆπ SAS 10 Corresponding angles of congruent triangles are β π΅π΄π β β πΆπ΄π congruent. 11 Μ Μ Μ Μ π΄π bisects β π΅π΄πΆ Definition of Angle Bisector SUMMARY Todayβs lesson used properties of isosceles triangles to complete proofs. MOD1 L23 4