When constructing a line segment, we use a compass and straightedge to first draw a ray or line and then a point that will serve as an endpoint of the new segment. Next, we measure the given segment with a compass and make a mark with the pencil end. Without changing the spacing of the compass, place the sharp end of the compass on the point drawn on the new line/ray, and make a mark on the line/ray. This is our line segment.
The rules of construction say, you can only have two tools at your disposal. The first is a compass and I know what you're probably thinking at home that does not look anything like my compass Mr. McCall. But it basically accomplishes the same thing, I can control the radius just by shifting the marker part. And the second piece is a straight edge, again the straight edge means that you have no markings on your straight edge. Now mine it does have markings so I guess we'd call a ruler but in tradition Greek construction we won't be able to have any markings on here. The first thing we're going to talk about is how to duplicate a line segment. If I looked at this line segment ab and I wanted to congruent line segment the first step is I take my straight edge and I'm going to draw a ray on to which I can construct my new line segment. So here we have a ray, I'm going to name my end point c since I already have a and b. And when you duplicate a line segment all you're really doing is measuring it with your compass. So you're going to take your sharp end of your compass or the rubber end for me and you're going to put it on the end point a. And then you're going to stretch your compass until it measures exactly the distance of your line segment. And the way that you verify that, is by making a little mark on the end of your line segment. So I'm going to mark right there on b. Now on your quiz do not erase that mark, that's what your teacher is going to look for to make sure you're doing it correctly. So then you're going to take without changing your compass the sharp end on the point c and you're going to create that same mark. So basically by keeping this rigid we measured ab and we just duplicated it down on the bottom and you can label this point d. So that's the essence of duplicating congruent line segments.