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
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.