Answer
a. 𝑥 = 75 degrees
b. 𝑥 = 48 degrees
Work Step by Step
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
