The locus of a point which moves so that it is an equal distance from two points, A and B, is the perpendicular bisector of the line joining A and B.
Perpendicular means at right angles to.
Bisector means cuts in half.
To construct this locus, you do the following (try this yourself on a piece of paper):
Perpendicular bisector of a line segment
Practise this construction until you can do it without looking at the instructions. You don't need to label the points with letters each time - they are just there as a guide.
To bisect an angle, you do the following:
Bisecting an angle
Try this yourself on a piece of paper.
Remember: if you are asked to do a construction in an exam, do not rub out your construction lines.
The loci of a point which moves so that it is the same distance from lines AOB and COD, are the lines which bisect angles AOC, AOD, DOB and BOC: