In triangle ABC, AB = AC,& angle A=20degrees.M is a point on Ab such that angle MCB=50degrees and N is a point

Question

In triangle ABC, AB = AC,& angle A=20degrees.M is a point on Ab such that angle MCB=50degrees and N is a point on AC such that angle NBC is 60 degrees. Calculate angle BNM

Answer 1

Given:
BAC = 20. AB = AC. MCB = 50. NBC = 60.

Therefore ABC is an isoceles triangle, which means CBA = BCA = (180 - 20) / 2 = 80.

It also means than NM is NOT parallel to BC.

Thus:
MCN = BCA - MCB = 80 - 50 = 30
NBM = CBA - NBC = 80 - 60 = 20
CMB = 180 - NBM - NBC - MCB = 180 - 20 - 60 - 50 = 50
BNC = 180 - MCN - MCB - NBC = 180 - 30 - 50 - 60 = 40

Let X represent point where BN crosses CM
BXC = NXM = 180 - NBC - MCB = 180 - 60 - 50 = 70
And thus
BXM = CXN = 180 - BXC = 180 - 70 = 110

MXN + CMN + BNM = 180
CMN + BNM = 180 - 70 = 110

MN = XN
Therefore CMN = MXN = 70
Therefore
BNM = 180 - CMN - MXN = 180 - 70 - 70 = 40

Answer 2

60 degrees

Answer 3

It is 60 degrees because it is the same as NBC, structure wise.