Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
First, construct a 90º angle, and then bisect it to create a 45º angle.
Starting with a ray, first create a perpendicular bisector. Remember that perpendicular lines create 90º angles. Then, placing the compass at the vertex of the angle, swing an arc, intersecting both sides. From each point of intersection, swing arcs of equal size. Using the straightedge, connect the vertex with the point at which these two arcs intersect. This creates two angles of equal size. Therefore, these are two angles that are half of 90º, so each is 45º.
A common application of angle bisectors is constructing a 45 degree angle. But how are you going to start by constructing a 45 degree angle? Well, if you think about bisecting, if you doubled 45, that’s 90 and you know how to make a 90 angle. So our first step is to make a 90 degrees angle. Once we’ve done that, we can bisect that angle. So let’s get started.
The first step which I’ve already done is to draw a ray. You need some place to start. But how are we going to create a 90 degree angle? Well we know that if you bisect the line segment, you’ve created its perpendicular bisector.
So first let’s make a line segment, so I’m, going to grab my compass and actually you can just do this with your pencil. Just draw on a random endpoint and we’re going to bisect that line segment. So you’re going to grab your compass and you’re going to swing an arc from one of those endpoints. You’re going to come over here and you’re not changing your compass and you swing another arc. Okay.
And it looks like my compass got off the board there for a little bit but it doesn’t really matter because I have my two points of intersections. So now I’m going to connect them with my straightedge. Okay, so we have created the perpendicular bisector of that line segment. Since it’s perpendicular I know that this is 90 degrees.
Now half of 90 degrees is 45. So let’s bisect this angle. So you’re going to take your compass and you’re going to swing another arc and let’s change its distance a little bit. Swing another arc, so that this angle right here has two places of intersection. From those two places we’re going to swing two more arcs. So I’m going to swing an arc right off there and I’m going to come over here and I’m going to swing another arc. So where those two have intersected, that has to be on the perpendicular bisector.
So I’m going to take my straightedge and I’m going to connect my vertex with that point of intersection. And we have created a 45 degree angle. Now sometimes on a test you might have a true and false question is it possible to construct a 22.5 degree angle, or your teacher might ask you to do that. Once you’ve created your 90 degrees angle, if you bisect it once, you get 45 degrees. If you bisect 45 degrees you’d end up with 22.5. So keep that in mind next time you see a 22.5 degree angle on your constructions test.