Geometry Chapter 5.1 Perpendiculars and Bisectors:
I) Vocabulary:
A) Perpendicular Bisectors - a segment, ray, line or plane that is perpendicular to a segment at its midpoint.
B) Equidistant - a point is equidistant from two points if its distance from each point is the same.
C) Distance from a point to a line - is defined as the length of the perpendicular segment from the point to the line.
D) Equidistant from the two lines - when a point is the same distance from one line as it is from another line.
II) Theorems:
A) Perpendicular Bisector Theorem - if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Example: If line CP is the perpendicular bisector of line segment AB, then CA = CB.
B) Converse of the Perpendicular Bisector Theorem - if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Example: If DA = DB, then point D lies on the perpendicular bisector of line segment AB.
C) Angle Bisector Theorem - if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Example: if measure of angle BAD = measure of angle CAD, then BD = DC.
D) Converse of the Angle Bisector theorem - if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.
Example: If DB = DC, then measure of angle BAD = measure of angle CAD
