
Method 1:
Using the striaght edge, draw two intersecting lines, forming an angle. Label the vertex as point A.
Draw a large arc, centered at A, and cross over both lines. This creates two new points, B and C.
Bisect the angle. This creates point D.
Center the compass at B, and draw a circle so that it intersects D. This will create another point on the large arc, E.
Now, draw a straight line from E to A. The result: angle EAC is perfectly trisected. It may not be the original angle you started off with, but so what? 

Method 2:
Draw a straight line.
Draw another straight line perpendicular to the first.
Center the compass at the intersection, and draw an arc 3/4 of the way around, starting on one leg and ending at another (see diagram).
Congratulations! You have just trisected a 270degree angle. 

Method 3:
Draw a straight line using the straight edge.
Keeping the straight edge exactly in place, trace two more lines directly over the original line (see diagram).
Congratulations! You have just trisected a 0degree angle. 
Method 4:
Okay, all kidding aside, it is time to seriously impress your friends. First of all, it is important that your straight edge has absolutely no hash marks on it whatsoever. Remember, you are about to successfully trisect an angle using nothing more than the aid of a simple compass and a blank straight edge. Using a ruler (with markings) is cheating.
Have someone else draw the original angle. Bisect the angle, followed by bisecting the halfangles into quarter angles, and continue to draw arcs and lines using the compass and straight edge.
Now the tricky part: when no one is looking, whip out your walletsized protractor, slap in on the drawing, quickly measure the angle, and mentally divide the angle by 3 as you draw hash marks on the paper.
***Hide the protractor!*** This is very IMPORTANT if you want to continue the illusion of trisecting an angle using only a compass and straight edge. If anyone else sees the protracter, the demonstration will not be as convincing.
Continue drawing more arcs and lines, but this time make sure that they intersect the two hash marks. Now draw lines from the vertex to the hash marks, and you have trisected the angle. 