How do you construct congruent segments, segment bisectors, angles, and angle bisectors using tools such as a compass and straightedge?

Constructing angles is a bit complicated. But remember you can only construct angles divisible by 15 with a compass and a straightedge (e.g. 15,30, 45,60 and so on..) Here's the link- 
http://www.youtube.com/watch?v=Pn01AsuDFnc 

They are all accurate if you draw them taking all the precautions. It might seem difficult in the starting but once you get the hold of it you will be able to make them all properly. 

I hope it helped