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
A 30º angle can be constructed by first constructing a 60º angle, then bisecting it to create two 30º angles.
First, on a ray, set the compass at one endpoint and swing an arc from that endpoint. Placing the compass at the point where this arc intersects the ray, draw another arc. The point at which these two points intersect is an equal distance from both endpoints, so connecting this point to the two endpoints would create an equilateral triangle. Remember that the angles of an equilateral triangle are all 60º. So, by connecting the new point to one of the endpoints, there is now a 60º angle.
Changing the size of your compass, draw an arc from the vertex of the angle that intersects the angle at two points. Then, draw two more arcs at each of these points of intersection. Where these arcs intersect is a line equidistant from both sides of the angle. By drawing a ray from the vertex through this new point of intersection, the 60º angle is bisected, creating two 30º angles.
A common application of constructing angle bisectors is creating various types of angles. In this problem we’re going to talk about constructing a 30 degree angle. Remember the only two things you can use are a compass and a straightedge. So what’s our game plan going to be?
Well, step one is going to be, construct a 60 degree angle. We’re going to do that by constructing three congruent line segments. Once we’ve constructed a 60 degree see if we divide that in half, we’re going to get 30. So the second step is going to be, bisect the angle that we’ve created. So let’s do it.
We’re going to grab our compass, and it doesn’t really matter what you set your compass at as long as you’re consistent and I’m going to take an endpoint. So the first step I’ve already drawn a ray. So at home if you’re following along, draw a ray. And I’m going to swing an arc from that endpoint.
Keeping my compass the same, I’m going to go over to this point of intersection and I’m going to swing another arc. So this point right here is the same distance from both these endpoints, which would create an equilateral triangle. Now as you remember, equilateral triangles have angles that are all 60 degrees.
So if I connect these two points right here, then I’ve constructed a 60 degree angle. So if we’re following along our game plan, we’re done with constructing a 60 degree angle now we need to bisect it. So take out your compass again, and I’m going to change my compass a little bit. I’m going to swing an arc from the vertex so that it intersects my angle in two places.
Now you can change your compass setting but you don’t have to, and from both of these points right here you’re going to swing two more arcs. You want to see where they intersect, because on that point it will be equidistant from your rays.
So, I’m going to grab a red marker so you can tell the difference here. And I’m going to connect that point of intersection with my vertex, which will create a 30 degree angle. So the key is coming up with a game plan. We know we construct, can construct a 60 degree angle and if you divide 60 in half, we would have 30 degrees.
So if you’re asked on a test to construct a 15 degree angle, what would you do? Well you would do the exact same procedure, except, now you would bisect that 30 degree angle.