Using Angle Bisectors of Triangles

As we move through this lesson keep in mind the things that you learned about perpendicular bisectors. You will notice similarities and differences.

Here's a quick at a glance to help you out.

Angle Bisector: an angle bisector is a ray that divides an angle into two congruent adjacent angles.  $\overline{BD}$ is the angle bisector.

Distance from a point to a line: The length of the perpendicular segment from the point to the line. (red dashed line)

Angle Bisector Theorem:

If a point is on the bisector of an angle, it is equidistant from the sides of the angle.

Note that this theorem allows us to find side length of the perpendicular lines if we know the angles are congruent.

Converse of the Angle Bisector Theorem:

If a point is on the interior of an angle and is equidistant from the two sides of an angle, it is on the angle bisector.

Note that this theorem allows us to find angle measures of the bisected angle if we know the perpendicular lines are congruent.

Solution: 45.

Explanation:  Since  $\overline{CB}\perp\overline{CD}$  and  $\overline{BE}\perp\overline{DE}$  and  $CB=BE=7$ ,  $\overrightarrow{DB}$  bisects  $\angle CDE$  by the Converse of the Angle Bisector Theorem. So,  $m\angle BDC=m\angle BDE\ =45\degree.$  

EF = 87 by the Angle Bisector Theorem.

Ray CE bisects angle BCD by the converse of the angle bisector theorem, angle BCE is congruent to angle DCE, so:

a+26=2a

26 = a

By the angle bisector theorem, we know that GH=HE.

u+26=2u

26=u

The point of concurrency of angle bisectors of a triangle is called the incenter.

P is equidistant from each side of the triangle.

The incenter always lies inside the triangle.

The incenter is equidistant from all three sides of the triangle.

It is the center of a circle inscribed within the triangle that touches all three sides.