Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle
The Converse of the Angle Bisector Theorem states that:
If a line segment splits an angle of a triangle into two congruent parts, then it must be the bisector of that angle.
In other words, if a line segment from a vertex of a triangle to the opposite side splits the angle at that vertex into two equal angles, then that line segment must be the angle bisector.
This converse theorem is useful in geometry proofs where we are already given information about the angle bisectors and need to prove that certain angles are congruent or triangles are similar.
For example, if we are given that the angle bisectors of a triangle are concurrent (meet at a point), we can use the converse of the Angle Bisector Theorem to prove that the three angles of the triangle are congruent. Another example is if we are given that two triangles have the same angle bisectors, we can use the converse of the Angle Bisector Theorem to prove that these triangles are similar.
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