## How To Bisect An Angle

Constructing an angle bisector with only a drawing compass, pencil, and straightedge is a wonderfully satisfying geometric construction. It is quick and easy, requiring just a single setting for your compass.

The angle bisector is the line, line segment or ray that cuts a given angle exactly in half. If you had a $60°$ angle, the angle bisector would produce two $30°$ angles. If your angle were open to $138°$, the angle bisector would give you two $68°$ angles.

[insert drawing illustrating an angle bisected]

Imagine getting a slice of apple pie and thinking it is too big. If you cut the slice in half from the outer edge to its inner point, your cutting line is an angle bisector.

[insert drawing or animation of angle bisector through a pie slice]

### How To Construct an Angle Bisector Given An Angle

If you are given an existing angle formed by rays (or line segments or even intersecting lines), you can find the angle bisector by placing the needle arm of the compass on the vertex.

Open your drawing compass to a setting that takes the pencil end very far along each ray. Sweep the compass to draw arcs across the two rays, to establish points on each ray that are equidistant from the vertex.

Though you can change the compass at this time, you have no reason to make it wider or narrower.

Relocate the needle to the intersection of your little arcs and each ray, swinging the compass to draw arcs within the general area inside the angle (but past the drawn rays, not closer to the vertex).

Making that same move twice will produce intersecting arcs in an area beyond the drawn rays but still within the angle's open "jaws." Recall that rays extend forever in one direction, so all the space bounded by the two rays is in the interior of the angle.

Use that point of intersection of the two arcs to line up your straightedge with the vertex. The line, line segment or ray you draw from the vertex through the intersecting arcs is an angle bisector. You are done!

[insert drawing or animation of steps]

### Angle Bisector Construction

If no angle exists, you get the fun of constructing the angle. Go right ahead. Construct an angle using your straightedge. Set down a point to be the vertex, then extend two rays from that common vertex, using the straightedge.

You need only the drawing compass, not a protractor since you do not need to know the actual measurement of the angle to bisect it.

Label the vertex with a lettered point. Use your compass to sweep equal-length arcs from the vertex along each ray. Where the arcs intersect the rays, label those points with letters, too. You now have your two intersecting rays and three identified points.

As before, use the newly labeled points on the rays as the location for the needle arm of the compass and swing arcs outward, past the rays but still within the interior of the angle. And, just as before, the intersecting arcs are your exact spot for placing the straightedge and connecting it to the vertex. Angle bisected!

### Next Lesson: Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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