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GEOMETRY Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM Name ____________________________ M1 Date _______________ Lesson 23: Base Angles of Isosceles Triangles Today we examine a geometry fact that we already accept to be true. We are going to prove this known fact by using SAS triangle congruence criteria. Here is isosceles triangle ABC. We accept that an isosceles triangle, which has (at least) two congruent sides, also has congruent base angles. Label the congruent angles (base angles) in figure 1. Label the vertex angle in figure 1. The following is a proof which demonstrates that the base angles of an isosceles triangle are always congruent. Figure 1 Prove Base Angles of an Isosceles Triangle are Congruent: SAS Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵 = 𝐴𝐶 Prove: ∠𝐵 ≅ ∠𝐶 Statements Reasons 1. Isosceles Triangle 1. given 2. AD bisects BAD 2. auxiliary line 3. mBAD mCAD 3. angle bisectors divide angles into 2 equal measures 4. AD = AD 4. reflexive property 5. BAD CAD 5. SAS congruence 6. B C 6. corresponding angles of congruent triangle are congruent Lesson 23: Date: Construction: Draw the angle bisector AD of ∠𝐴, where 𝐷 is the intersection of the ̅̅̅̅ . The bisector and 𝐵𝐶 auxiliary line will be used for the SAS criteria. Base Angles of Isosceles Triangles 11/23/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.125 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. In proofs, we can state that “base angles of an isosceles triangle are equal in measure” or that “base angles of an isosceles triangle are congruent” if we already have the isosceles triangle with the congruent sides stated in the given. Otherwise, if we do not know that we have an isosceles triangle already then we can state… “In a triangle, angles opposite equal sides are equal” as well as “In a triangle, sides opposite equal angles are equal.” Exercises 1. 2. 3. 4. Lesson 23: Date: Base Angles of Isosceles Triangles 11/23/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.126 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 5. 6. Additional Properties: *** Two angles that are equal in measure that form a linear pair are right angles. *** Two angles that are supplementary to the same angle or angles of equal measure are equal. Lesson 23: Date: Base Angles of Isosceles Triangles 11/23/15 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.127 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.