When constructing an angle, first swing an arc from the vertex of your angle. Then, swing a congruent arc from the new vertex. Return to the original angle; the drawn arc intersected the sides of the angle - measure this distance. On the new angle, place the sharp end of the compass on the intersection of the arc and ray and draw another arc. Draw a ray connecting the new vertex with the point of intersection.
One of the most basic instructions that you can perform is duplicating an angle. The reason why this is important is if you want to duplicate parallel lines but we're not going to talk about that just here. Here we have an angle and I'm going to duplicate it onto a ray. So first I need to draw a ray right over here and this ray will give us a spot to mark our constructions. So first step here is to take your compass and again my compass looks a little bit different than yours at home and you're going to swing an arc and here it doesn't matter how wide your marker is or how wide your compass is. So you're going to put the sharp end on the vertex and you're going to a swing an arc so that you have two points of intersection on your rays. Without changing your compass again, this is the key part you're going to come over here and you're going to swing an exact same arc. Now you've swung two arcs, how do you know how much this angle opens? We're basically going to measure that distance between these two intersections with your compass. So you're going to take the sharp edge and you're going to measure it with your compass. So once you have that distance, you're going to make a mark, now it doesn't matter where you make this mark you can make it up here as well. But this mark tells your teacher that you had not only swung this arc but you've also measured that distance. This is what he or she will be looking at on your quiz. You're going to come back over here and you're going to take the sharp edge of your compass and put it on the only intersection that you have. So you're going to put the sharp edge right here and you're going to swing that exact same mark so you know that this opening distance is the same as this distance right here. So now you can take your pen or my case a marker and you're going to connect that point of intersection with your vertex thereby duplicating your angle. So the key step here was swinging the exact same mark and not changing when you go from your original to your new image.