Question Video: Determining the Tools Required to Construct an Angle Bisector without a Protractor Mathematics

True or False: Using only a straightedge, you can construct an angle bisector.

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Video Transcript

True or False: Using only a straightedge, you can construct an angle bisector.

To determine whether this statement is true, we’ll need to review the steps required to construct an angle bisector. Let’s sketch ray 𝐴𝐵 and ray 𝐴𝐶 to form an arbitrary angle 𝐵𝐴𝐶. We recall that our very first step requires tracing a circle centered at 𝐴 that intersects the sides of the angle at two distinct points.

Now, we can’t construct a circle using just a straightedge. It will require a compass. We place the point of our compass on the vertex 𝐴. Now, we don’t have to trace the whole circle, just an arc that intersects ray 𝐴𝐵 at point 𝐷 and ray 𝐴𝐶 at point 𝐸. The next step also requires a compass in order to trace two circles of the same radius centered at 𝐷 and 𝐸. These circles will intersect at a point that we name point 𝐹.

It is only in the final step that we use our straightedge to sketch the line 𝐴𝐹 that bisects angle 𝐵𝐴𝐶. As we have demonstrated, we can use a straightedge to construct an angle bisector, but not only a straightedge. Without a compass, we cannot complete the construction. Thus, the given statement is false.

We note that we could alternatively bisect an angle using only a ruler by constructing an isosceles triangle. We would have to measure the sides then connect the midpoint of the base to the opposite vertex.