Question

How can I construct MT ,the angle bisector of M ,to cut LN at T

Answer 1

not sure how M can be both a point and an angle.

I suspect you meant

construct MT ,the bisector of LN at T

Now, there are many bisectors of LN that pass through T, the midpoint of LN.

You can find many examples online of how to construct a perpendicular bisector of a line segment. That will give you the midpoint T of LN

Then just draw the line MT

If I misread your problem, then maybe you can restate it more clearly. But it sounds like you will still have to construct either a line or angle bisector, for which google is your friend.