Pennsylvania Geometry - 2020 Edition
topic badge
7.01 Angle bisectors and perpendicular bisectors
Lesson

Angle bisectors

Recall that an angle bisector is a ray from the vertex of the angle that divides an angle into two congruent angles.

Exploration

Let's have a look at the applet below, which shows the constructed angle bisector of $\angle ABC$ABC. If we move point $P$P along the angle bisector, or change the size of $\angle ABC$ABC, what relationship is always true? Can we explain why?

From the applet we can see that point P is always the same distance from the sides of the angle, no matter where it is on the angle bisector or how large we make $\angle ABC$ABC.

Why does this happen? In the applet we can see two right triangles formed. We can prove these triangles are congruent by angle-angle-side congruence. Therefore, the distances, being parts of those triangles, are also always the same.

Angle bisector theorem and converse

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

If $\overrightarrow{AD}$AD bisects $\angle BAC$BAC and $\overline{DB}\perp\overline{AB}$DBAB and $\overline{DC}\perp\overline{AC}$DCAC then $DB=DC$DB=DC.
   

 

Converse of the angle bisector theorem - If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

If $BD=DC$BD=DC and $\overline{DB}\perp\overline{AB}$DBAB and $\overline{DC}\perp\overline{AC}$DCAC then $\overrightarrow{AD}$AD bisects $\angle BAC$BAC
   

We can prove both the theorem and its converse using congruent triangles.

 

Perpendicular bisectors

Recall that a segment bisector is a segment, line, or plane that intersects a segment at its midpoint. If a bisector is also perpendicular to the segment, it is called a perpendicular bisector.

Exploration

Let's have a look at the properties of a perpendicular bisector. Move points, $A$A, $B$B, and $P$P in the applet below. What relationships in the diagram are always true? Can you explain why?

The perpendicular bisector of a line segment is the set of all points that are equidistant from the segment's endpoints. Notice, that no matter where we move point $P$P, it is always the same distance from points $A$A and $B$B. This is a theorem that can be proven using congruent triangles.

Perpendicular bisector theorem and converse

Perpendicular bisector theorem - In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

If $\overleftrightarrow{CP}$CP is the $\perp$ bisector of $\overline{AB}$AB then $CA=CB$CA=CB.
   

 

Converse of the perpendicular bisector theorem - In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

If $DA=DB$DA=DB then $D$D lies on the $\perp$ bisector of $\overline{AB}$AB
   

The converse of the theorem is also true, and we can prove it using congruent triangles as well.

What is Mathspace

About Mathspace