Transcript We have guessed that when an angle of a triangle is

We have guessed that when an angle of a triangle
is bisected, the opposite side is divided into
segments that are proportional to the other sides.
This statement is merely a conjecture,
arrived at inductively. We need to reason
deductively in order to be assured that this
will always be the case, for all triangles.
•

Let BD be the angle
bisect or ofABC.
B
A
D
C
Construct CE parallel
to BD.
E
B

A
D
C
Ext endAB so that it intersects
F
wit h CE. Call this point of
intersect ion F.
E
B

A
D
C
• By the Basic
Proportionality
Theorem,
AB/BF=AD/DC.
F
E
B
A
D
C
Re call that ABD  DBC
by definition of angle
bisector.

F
E
B
A
D
C
Since BD CF, corresponding angles
F
ABD and AFC are congruent .
E
B

A
D
C
Also result ing fromBD CF is the fact
F
that alt ernate interior angles
DBC and
BCF are congruent .

E
B
A
D
C
Since AFC  ABD
ABD  DBC
DBC  BCF
F
E
then AFC  BCF
B
A
D

C
By the converse of the
F
isosceles triangle theorem,
BF  BC.
E
B
A

D
C
However, we stated earlier that
AB AD

. We can substitut e BC
BF DC
in for BF, and conclude
AB AD

.
BC DC
F
E
B
A
D

C
Using properties of proportions,
we can also
AB BC
AD DC
write

or

AD DC
AB BC.
