Theorems About Lines and Angles
Rule

Converse Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the segment.
Point C on the perpendicular bisector equidistant from endpoints A and B
Based on the characteristics of the diagram, the following relation holds true.

Proof

Converse Perpendicular Bisector Theorem

Consider and a point equidistant from and

Point C is equidistant from endpoints A and B
To prove that lies on the perpendicular bisector of it will be shown that the line perpendicular to through bisects If is the point of intersection between the line and the segment, it must be proven that
Draw a line perpendicular to segment AB
This line forms two right triangles that share a common leg Because all right angles are congruent, is congruent to Also, by the Reflexive Property of Congruence, is congruent to itself. Since is equal to is congruent to
By the Hypotenuse Leg Theorem, and are congruent triangles. Because corresponding parts of congruent figures are congruent, is congruent to
Segment AM and segment BM are congruent segments

Additionally, it was already known that and are perpendicular.

By the definition of a perpendicular bisector, is the perpendicular bisector of Therefore, lies on the perpendicular bisector of

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