1. Perpendicular and Angle Bisectors Perpendicular Bisector – A line, segment, or ray that passes through the midpoint of a side of a triangle and is perpendicular to that side. Theorem 5.1 Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Theorem 5.2 Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

2. Perpendicular and Angle Bisectors Concurrent Lines – Three or more lines that intersect at a common point. Point of Concurrency – The point of intersection of the concurrent lines. Circumcenter – The point of concurrency of the perpendicular bisectors of a triangle. Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle.

3. Perpendicular and Angle Bisectors Angle Bisector – A segment, line, or ray that passes through the vertex of an angle and divides that angle into two congruent pieces. Theorem 5.4 Any point on the angle bisector is equidistant from the sides of the angle. Theorem 5.5 Any point equidistant from the sides of an angle lies on the angle bisector.

4. Perpendicular and Angle Bisectors Incenter – The point of concurrency of the angle bisectors of a triangle. Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle.

5. Example 1-1b Prove: Given:.

6. Example 1-1b Proof: Statements. Reasons 1. Given 2. Angle Sum Theorem 3. Substitution 4. Subtraction Property 5. Definition of angle bisector 6. Angle Sum Theorem 7. Substitution 8. Subtraction Property 1. 2. 3. 4. 5. 6. 7. 8. 1. Given