Warm Up– on scratch paper  

Warm Up #2 How could you go about solving for x in the triangles below?

Angle Bisectors of Triangles Advanced Geometry

Distance Between a Point and a Line The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.

Equidistant When a point is the same distance from one line as it is from another line, then the point is equidistant from the two lines.

ANGLE BISECTOR THEOREM If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

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.

Angle Bisector of a Triangle An angle bisector of a triangle is a bisector of an angle of the triangle.

INCENTER OF THE TRIANGLE The point of concurrency of the angle bisectors is called the incenter of the triangle.

CONCURRENCY OF ANGLE BISECTORS OF A TRIANGLE The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangles

Incenter always lies inside the triangle!

Example What segments are congruent? If AY = 15 and AP = 17, then find PY and PX.

How to remember points of concurrency? ABI loves to eat Peanut Butter Cups. Peanut Butter Cups: Perpendicular Bisectors – Circumcenter ABI: Angle Bisectors – Incenter

Remember! Circumcenter is equidistant to the verticies! Can be inside, on, or outside the triangle Incenter is equidistant to the sides! Is always inside the triangle