Chapter 4 - Congruent Triangles - 4-5 Isosceles and Equilateral Triangles - Lesson Check - Page 253: 2

a. 𝑥 = 75 degrees b. 𝑥 = 48 degrees

a. ED = FE - given 𝑚∠E = 30 degrees - given 𝑚∠D = 𝑥 degrees - given If ED = FE, then ∠D = ∠F - Theorem 4-4 Converse of the Isosceles Triangle Theorem 𝑚∠F = 𝑥 degrees - Theorem 4-4 Converse of the Isosceles Triangle Theorem 𝑚∠E + 𝑚∠D + 𝑚∠F = 180 - Triangle Angle-Sum Theorem 30 + 𝑥 + 𝑥 = 180 - Substitution (30 + 𝑥 + 𝑥) - 30 = 180 - 30 - Subtract 30 from both sides 𝑥 + 𝑥 = 150 - Add like terms (𝑥) 2𝑥 = 150 2𝑥/2 = 150/2 - Divide both sides by 2 𝑥 = 75 degrees b. LN = MN - given ON = ON - Reflexive Property of Congruence 𝑚∠L = 42 degrees - given 𝑚∠ONM = 𝑥 degrees - given 𝑚∠ONM = 𝑚∠ONL - given ON bisects ∠LNM - given △ONL ≅ △ONM - Postulate 4-2 Side-Angle-Side (SAS) Postulate 𝑚∠NOL = 𝑚∠NOM = 90 degrees - Theorem 4-5 Vertex Angle of Isosceles 𝑚∠L + 𝑚∠NOL + 𝑚∠ONL = 180 - Triangle Angle-Sum Theorem 42 + 90 + 𝑥 = 180 - Substitution 132 + 𝑥 = 180 132 + 𝑥 - 132 = 180 - 132 - Subtract 132 from both sides 𝑥 = 48 degrees