1. Warm Up– on scratch paper

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

3. Angle Bisectors of Triangles
Advanced Geometry

4. 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.

5. 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.

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

7. 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.

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

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

10. 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

11. Incenter always lies inside the triangle!

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

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

14. 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