Download Angle Bisector

5.6
ESSENTIAL OBJECTIVES:
 Use angle bisectors and perpendicular
bisectors in solving
triangles.
How do you apply the
angle bisector and
perpendicular bisector
concept in solving for
missing parts of a
triangle?
Angle Bisector

A segment that bisects, or splits an angle
into 2 congruent angles.
Angle Bisector Theorem

If a point is on the bisector of an angle,
then it is equidistant from the two sides of
the angle.
Angle Bisector Theorem

If a point is on the bisector of an angle,
then it is equidistant from the two sides of
the angle.
Example 1
Use the Angle Bisector Theorem
UW bisects TUV.
∆TWU  ∆VWU.
SOLUTION
Statements
Reasons
1. UW bisects TUV.
.
1. Given
3. WU  WU
4. WV  WT
5. ∆TWU  ∆VWU
3. Reflexive Prop. of Congruence
4. Angle Bisector Theorem
5. HL Congruence Theorem
.
Checkpoint
1.
Use Angle Bisectors and Perpendicular
Bisectors
Find FH.
Checkpoint
Find FH.
Use Angle Bisectors and Perpendicular
Bisectors
Checkpoint
Find FH.
Use Angle Bisectors and Perpendicular
Bisectors
Find FH.
Checkpoint
Use Angle Bisectors and Perpendicular
Bisectors
Checkpoint
2.
Use Angle Bisectors and Perpendicular
Bisectors
Find MK.
Checkpoint
Find MK.
Use Angle Bisectors and Perpendicular
Bisectors
Perpendicular Bisector

A segment that bisects, or splits an angle
into 2 congruent angles.
Checkpoint
3.
Use Angle Bisectors and Perpendicular
Bisectors
Find EF.
Checkpoint
Find EF.
Use Angle Bisectors and Perpendicular
Bisectors
Example 2
Use Perpendicular Bisectors
Use the diagram to find AB.
SOLUTION
In the diagram, AC is the
perpendicular bisector of DB.
8x = 5x +12
By the Perpendicular Bisector
Theorem, AB = AD.
3x = 12
3x 12
=
3
3
x=4
Subtract 5x from each side.
Divide each side by 3.
Simplify.
You are asked to find AB, not just the value of x.
ANSWER
AB = 8x = 8 · 4 = 32
Example 4
Use Intersecting Bisectors of a Triangle
A company plans to build a warehouse
that is equidistant from each of its three
stores, A, B, and C. Where should the
warehouse be built?
SOLUTION
Think of the stores as the vertices of a triangle. The
point where the perpendicular bisectors intersect
will be equidistant from each store.
1. Trace the location of the stores on
a piece of paper. Connect the
points of the locations to form
∆ABC.
Example 4
Use Intersecting Bisectors of a Triangle
2. Draw the perpendicular bisectors
of AB, BC, and CA. Label the
intersection of the bisectors P.
ANSWER
Because P is equidistant from each vertex
of ∆ABC, the warehouse should be built
near location P.
Review
1. Sketch the overlapping triangles separately. Mark all
congruent angles and sides. Then tell what theorem or
postulate you can use to show that the triangles are
congruent.
KL  NM, KLM  NML
ANSWER
SAS Congruence Postulate
2. Use the information in the diagram to show that
the given statement is true.
HJ  KJ
ANSWER
It is given that GH  GK, and GJ  GJ by the Reflexive
Property of Congruence. ∆GHJ and ∆GKJ are right
triangles by definition. By the HL Congruence
Theorem, ∆GHJ  ∆GKJ. Therefore, HJ  KJ because
corresponding parts of congruent triangles are
congruent.

Hw 5.6A & 5.6B (4-9)