It is possible to draw two bisectors of a given angle.
Idea: An angle bisector is a ray from the vertex that cuts the angle into two equal parts.
1) Name the angle as \(\angle AOB\).
\(\angle AOB\)
2) If a ray \(OC\) bisects the angle, then the two small angles are equal:
\(\angle AOC = \angle COB\)
3) Each of these equals half of the original angle:
\(\angle AOC = \angle COB = \tfrac{1}{2}\,\angle AOB\)
4) Suppose, for the sake of checking, there is another different bisector ray \(OD\).
Then we would also have:
\(\angle AOD = \angle DOB = \tfrac{1}{2}\,\angle AOB\)
5) But if \(OC\) and \(OD\) are different rays, one of the angles \(\angle AOC\) or \(\angle AOD\) must be smaller than the other (because the rays point in different directions).
That cannot happen if both are exactly \(\tfrac{1}{2}\,\angle AOB\). They can only be equal when the rays are the same line.
6) Therefore, an angle has only one bisector.
Conclusion: The statement “It is possible to draw two bisectors of a given angle” is false.