Answer
a. π₯ = 75 degrees
b. π₯ = 48 degrees
Work Step by Step
a.
ED = FE - given
πβ E = 30 degrees - given
πβ D = π₯ degrees - given
If ED = FE, then β D = β F - Theorem 4-4 Converse of the Isosceles Triangle Theorem
πβ F = π₯ degrees - Theorem 4-4 Converse of the Isosceles Triangle Theorem
πβ E + πβ D + πβ F = 180 - Triangle Angle-Sum Theorem
30 + π₯ + π₯ = 180 - Substitution
(30 + π₯ + π₯) - 30 = 180 - 30 - Subtract 30 from both sides
π₯ + π₯ = 150 - Add like terms (π₯)
2π₯ = 150
2π₯/2 = 150/2 - Divide both sides by 2
π₯ = 75 degrees
b.
LN = MN - given
ON = ON - Reflexive Property of Congruence
πβ L = 42 degrees - given
πβ ONM = π₯ degrees - given
πβ ONM = πβ ONL - given
ON bisects β LNM - given
β³ONL β
β³ONM - Postulate 4-2 Side-Angle-Side (SAS) Postulate
πβ NOL = πβ NOM = 90 degrees - Theorem 4-5 Vertex Angle of Isosceles
πβ L + πβ NOL + πβ ONL = 180 - Triangle Angle-Sum Theorem
42 + 90 + π₯ = 180 - Substitution
132 + π₯ = 180
132 + π₯ - 132 = 180 - 132 - Subtract 132 from both sides
π₯ = 48 degrees