**Objective: **Apply
the definitions of the median and the altitude of a triangle and
the perpendicular bisector of a segment. State and apply related
theorems.

**Definitions:**

**Perpendicular Bisector Theorem:**

If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

**Converse of the Perpendicular Bisector Theorem:**

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

**Angle Bisector Equidistant Theorem:**

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

**Converse of the Angle Bisector Equidistant
Theorem:**

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

**Exercises:**

Triangle RST is needed for Exercises1-5

1. If K is the midpoint of ST, then RK is called a(n)_______ of triangle RST.

2. If RK is perpendicular to ST, then RK is called a(n) ________ of triangle RST.

3. If K is the midpoint of ST and RK perpendicular ST, then RK is called a(n)

________ of ST.

4. If RK is both an altitude and a median of Triangle RST, then:

**a.** Triangle RSK
congruent Triangle RTK by __________.

**b. **Triangle RST
is a(n) _________ triangle.

5. If R is on the perpendicular bisector of ST, then R is equidistant from _______ and _________. Thus ______=_______.

6. Prove the Perpendicular Bisector Theorem.

**Given: **The line
through point X is the perpendicular bisector of BC; A is also
on this line.

**Prove:** AB=AC

7. Prove the Converse of the Perpendicular Bisector Theorem.

**Given: **AB=AC

**Prove:** A is on
the perpendicular bisector of BC.

**Plan for proof**:
The perpendicular bisector of BC must contain the midpoint of
BC and be perpendicular to BC. Draw an auxiliary line containing
A that has one of these properties and prove that it has the other
property as well. For example, first draw a segment from A to
the midpoint X of BC. You can show that AX is perpendicular to
BC if you can show that Angle 1 congruent to Angle 2. Since these
angles are corresponding parts of two triangles, first show that
triangle AXB is congruent to triangle AXC.

8. Prove the Angle Bisector Equidistant Theorem.

**Given: **BZ bisects
Angle ABC; P lies on BZ; PX is perpendicular to BA; PY is perpendicular
to BC

**Prove: **PX=PY

9. Prove the converse of the Angle Bisector Equidistant Theorem.

**Given: **PX perpendicular
to BA; PY perpendicular to BC; PX=PY

**Prove: **Ray BP
bisects Angle ABC