Final answer:
The Converse of the Perpendicular Bisector Theorem is used in the proof of the SSS Triangle Congruence Theorem to show that if a point is equidistant from the endpoints of a segment, it lies on the bisector of the segment, which helps in proving that corresponding angles are equal.
Step-by-step explanation:
The SSS (Side-Side-Side) Triangle Congruence Theorem states that if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. During the proof of this theorem, the Converse of the Perpendicular Bisector Theorem is used. This converse theorem states that if a point is equidistant from the endpoints of a segment, then it lies on the bisector of that segment.
When proving the SSS Congruence Theorem, if you assume that you have two triangles, ∆ABC and ∆DEF, with sides AB = DE, BC = EF, and AC = DF, and you draw the perpendicular bisector of one side of one triangle, it will also bisect the corresponding side of the other triangle, given the sides are equal in length.
By the Converse of the Perpendicular Bisector Theorem, the vertices opposite the bisected side in each triangle are equidistant to the endpoints of the bisected side. This serves as a step in proving the two triangles are congruent, by showing that corresponding angles are equal and thus by the ASA (Angle-Side-Angle) postulate, the triangles are congruent.