Question

How to construct angle 150.5

Answer 1

It is impossible if you don't have any tools. You can construct angle 150 by adding together 90 and 60, both of which can be constructed without any special tools (you would need a compass and a ruler).
To construct angle 150.5 you would need a protractor. A protractor is a device that allows you to construct angles and measure them. 

Answer 2

The correct answer is:

You must use a protractor.

Draw the vertex of your angle, and a line to use as the edge of the angle.  Place the center of your protractor over the vertex of the angle, and line the bottom of the protractor up with the edge of the angle.

Find the 0° mark at the edge of the angle.  Follow the numbers along the protractor from this 0 to 150.5; make a mark above this number.

Move the protractor; use a straightedge to connect the mark with the vertex of your angle.