Geometric constructions: congruent angles | Congruence | High school geometry | Khan Academy

What we're going to do in this video is learn to construct congruent angles. And we're going to do it with, of course, a pen or a pencil. Here, I'm going to use a ruler as a straight edge, and then I'm going to use a tool known as a compass, which looks a little bit fancy. But what it allows us to do, and we'll apply using it in a little bit, is it allows us to draw perfect circles or arcs of a given radius. You pivot on one point here, and then you use your pen or your pencil to trace out the arc or the circle.

So let's just start with this angle right over here, and I'm going to construct an angle that is congruent to it. So let me make the vertex of my second angle right over there, and then let me draw one of the rays that originates at that vertex. I'm going to put this angle in a different orientation just to show that they don't even have to have the same orientation. So it's going to look something like that; that's one of the rays. But then we have to figure out where to put the other ray so that the two angles are congruent, and this is where our compass is going to be really useful.

So what I'm going to do is put the pivot point of the compass right at the vertex of the first angle, and I'm going to draw out an arc like this. What's useful about the compass is you can make—you can keep the radius constant, and you can see it intersects our first two rays at points. Let's just call this B and C, and I could call this point A right over here.

Now that I have my compass with the exact right radius right now, let me draw that right over here. But this alone won't allow us to draw the angle just yet. So let me draw it like this, and that is pretty good. Let's call this point right over here D; I know I'll call this one E. I want to figure out where to put my third point F so I can define ray EF so that these two angles are congruent.

What I can do is take my compass again and get a clear sense of the distance between C and B by adjusting my compass. So one point is on C, and my pencil is on B; so I have to get this right. So I have this distance right over here. I know this distance, and I've adjusted my compass accordingly so I can get that same distance right over there.

You can now imagine where I'm going to draw that second ray. That second ray, if I put point F right over here, my second ray—I can just draw between starting at point E right over here, going through point F. I could draw that a little bit neater, so it would look like that. My second ray—ignore that first little line I drew. I'm using a pen, which I don't recommend for you to do it; I'm doing it so that you can see it on this video.

Now, how do we know that this angle is now congruent to this angle right over here? One way to do it is to think about triangle BAC, triangle BAC and triangle—let's just call it DFE. So this triangle right over here—when we drew that first arc, we know that the distance between A and C is equivalent to the distance between A and B, and we kept the compass radius the same.

So we know that's also the distance between E and F and the distance between E and D. Then the second time when we adjusted our compass radius, we now know that the distance between B and C is the same as the distance between F and D or the length of BC is the same as the length of FD.

So it's very clear that we have congruent triangles. All of the three sides have the same measure, and therefore the corresponding angles must be congruent as well.