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Duplicating an Angle - Problem 1
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It is possible to draw an angle twice the size of a given angle by using a compass and a straightedge. First, draw a ray, creating the vertex of the new angle. Then, using the compass, swing an arc through the original angle, and then swing that same arc, holding the end of the compass at one endpoint of the ray. With the compass, measure the intersection of the arc through the original angle. Place the compass on the intersection of the arc with the ray--this intersection is the point through which the second side of the angle can be drawn.

Repeat the process, using the new ray as the starting point to duplicate the angle. The resulting angle is twice the size of the original angle.

One application of duplicating angles is something like this problem where you’re being asked to duplicate this angle twice. That’s what they mean by 2ABC. They’re saying, take this angle and create one that’s twice as big. So to do this we’re going to need to start off by drawing a ray onto which we can duplicate.

So I’m going to come down here and I’m going to draw a ray. So this will give us our vertex of our new angle. The next step is to take your compass and we’re going to do all the steps of duplicating an angle.

So we’re going to start off by swinging an arc from our vertex coming down here, and our new vertex and swinging the exact same mark. And then we have to measure this opening’s distance. So we come up here and I’m going to measure, sharp end on one side and then my marker on the other side of that intersection. So now I can duplicate this angle by putting the sharp end on this intersection, so now I know that this will create our angle ABC.

But we’re not going to be done because we’ve only duplicated it once. So I’m going to come up here and we’ve now created 1ABC, so we have to do the process all over again. And just to show you that it doesn’t matter how large your arc is here, I’m going to choose a different width.

So I’m going to come up here and I’m going to swing another arc, notice that it’s larger than the other one and I’m going to come up here and I’m, going to swing an arc. Now why did I ignore this ray down here? Because I know that my new angle is going to go up here, so I’m not even concerned about this bottom ray. I’m not going to use it.

So now I need to measure that opening. So I’m going to put the sharp end on one side of that intersection and I’m going to try and find the other, okay? Again don’t erase your marks and I’m going to come right here. Sharp end on this intersection and I know my intersection over here is going to be right there. So now I can connect this vertex with that point of the intersection with my straightedge.

And what I’m going to do is I’m going to say that this is point DEF. So create 2ABC, I’m going to say that’s going to be equal to the angle of D, E and F. All you’re doing here is duplicating your angle twice, and you’re ignoring the bottom ray when you make your second angle.

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