To determine which of the statements are true, we will analyze each statement individually:
1. The perpendicular bisector of a segment is equidistant from the endpoints of that segment.
This statement is true. The perpendicular bisector of a segment is a line that divides the segment into two equal parts at a 90-degree angle. By definition, it is equidistant from the endpoints of the segment.
2. A perpendicular bisector can intersect the segment it bisects at any point.
This statement is true. A perpendicular bisector can intersect the segment it bisects at any point along its length. As long as the line meets the criteria of bisecting the segment and being perpendicular to it, it can intersect at any location.
3. An angle bisector is equidistant from the sides of the angle it is bisecting.
This statement is false. An angle bisector divides an angle into two equal angles. While the angle bisector does pass through the vertex of the angle, it is not necessarily equidistant from the sides of the angle. The distance from the angle bisector to the sides of the angle depends on the specific measurements of the angle.
4. An angle bisector creates two congruent vertical angles.
This statement is false. An angle bisector creates two congruent angles, but they are not necessarily vertical angles. Vertical angles are formed by the intersection of two lines, and an angle bisector divides an angle into two smaller angles.
Therefore, the true statements are:
- The perpendicular bisector of a segment is equidistant from the endpoints of that segment.
- A perpendicular bisector can intersect the segment it bisects at any point.