SOLUTION: Angle LNM = Angle KNM and LN is perpendicular to JK. Find the measure of Angle JKN. <img src="https://i.ibb.co/60kg1rH/Untitled-drawing-8.png" height="200px">

Question 1205619: Angle LNM = Angle KNM and LN is perpendicular to JK.
Find the measure of Angle JKN.

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

It's strange that point J is omitted as a label in this diagram.
I'll assume that point J is where the 161 degree angle is located.

The exterior angle 161 is adjacent to some interior angle x.
161+x = 180
x = 180-161
x = 19
Therefore angle NJK = 19 degrees.

Extend segment LN so that it intersects with JK.
Let's mark point P as the intersection of LN and JK.
This forms right triangle JPN.
Refer to the diagram below.
In the previous paragraph we found that angle J of triangle JPN is 19 degrees.
The other acute angle must be 90-19 = 71 degrees.
This is the measure of angle PNJ.

Angles PNJ and LNM are congruent vertical angles.
angle LNM = 71 and so is angle KNM due to the given info.

Focus on the angles: LNM, MNK, KNP
They glue together to form a straight line.
(angleLNM) + (angleMNK) + (angleKNP) = 180
(71) + (71) + (y) = 180
142+y = 180
y = 180-142
y = 38
angleKNP = 38 degrees

The last thing to focus on is right triangle PNK.
The conclusion of the last paragraph tells us that angle N = 38, so the other acute angle must be 90-38 = 52 degrees which is the final answer

I'll leave verification for the student to do.

RELATED QUESTIONS
angle KNM = 87 degrees angle LNM= (2x-8) degrees angle KNL = (X + 50) degrees What... (answered by )

Hello, Could you please help solve this problem? It doesn't specify that segment AB and... (answered by Alan3354)

Could you please help with finding angle DEF? (answered by greenestamps)
In the diagram below, find the measure of Angle BFD. Diagram:... (answered by ikleyn)
In the diagram, find the measure of Angle ∠BFD.... (answered by math_tutor2020)
In triangle ABC, DE is parallel to BC and AD = DE = BE. If BE is the angle bisector of... (answered by greenestamps)
The center of the circle is O, angle TQR = 46 degrees and angle QTO = 51 degrees. Find... (answered by greenestamps)
Find the sum of the following series: (answered by ikleyn,math_tutor2020)
In the diagram, AB is the diameter of the circle, and E is the centre. Find the measure... (answered by ikleyn)