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Section 5.3 ~ Angle Bisectors of Triangles
Topics in this lesson:
• incenter
• Angle Bisector Theorem
Objective: To understand what angle
bisectors are and what relationships exist
when they intersect inside a triangle.
angle bisector ­ a ray that divides an angle into two congruent angles
*This is a Section 1.4 definition.
incenter ­ the point of concurrency of the three angle bisectors in a triangle
*see picture on back side
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
*Remember that the distance from a point to a line has to be the perpendicular connection.
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.
*These theorems can be easily proven using congruent triangles and CPCTC.
Example 1
Find AD.
Example 2
Find the measure of CBE.
Example 3
Can you conclude that BD bisects ABC? Explain.
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 triangle.
*Be very careful not to confuse this with the Concurrency of Perpendicular Bisectors in 5.2!
Example 4
Point G is the incenter. Find BG.
Example 5
Find the value of x that makes N the incenter of the triangle.