Perpendicular and Angle Bisectors

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
6
Example 1-1b