Video Transcript
True or False: Using only a
straightedge, you can construct an angle bisector.
To determine whether this statement
is true, weโll need to review the steps required to construct an angle bisector. Letโs sketch ray ๐ด๐ต and ray ๐ด๐ถ
to form an arbitrary angle ๐ต๐ด๐ถ. We recall that our very first step
requires tracing a circle centered at ๐ด that intersects the sides of the angle at
two distinct points.
Now, we canโt construct a circle
using just a straightedge. It will require a compass. We place the point of our compass
on the vertex ๐ด. Now, we donโt have to trace the
whole circle, just an arc that intersects ray ๐ด๐ต at point ๐ท and ray ๐ด๐ถ at point
๐ธ. The next step also requires a
compass in order to trace two circles of the same radius centered at ๐ท and ๐ธ. These circles will intersect at a
point that we name point ๐น.
It is only in the final step that
we use our straightedge to sketch the line ๐ด๐น that bisects angle ๐ต๐ด๐ถ. 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.
