Einl Wand November 2014

proving the Angte Bisector Theorem and the Pythagorcan Theorem G.SRI.4 Worksheet t Guide 2
prove the Angte Bisector Theorem - An angle bisector of an artgleof a trlangledivides the opposite'side in,tow
segments that are proportional to the other two sides of the triangle.
\
:'\_s
t'xGiven:
MBC where BD ii
Prove:
+,
an angle bisector of
:
AB
'
lB'
\
o*
t
[,w- tt
fa-rdvl
8-Y1'u^L
nwJ Lrv'r-
€b
un$'t i+
a;Le 6
a-{- Po'rvt €
L
,t
16'ro'1h ft
'''1t'xz*s
"
tNL' lam's'rl
!l*wzw-'
-ta"l;,Lspli++^Ut
by
6c
o
f\
lt
enL
t0"-
'.-B
,'\
AD
BC= DC
iry
'
t1^q+
pa-'
e*A; lftb?
t{
/ ftey:/o6c
Levk'te
l+ Ats *
5o
&vv-' p'-"n<lu[
f ,>{-s
1
f
l
'nc-5
..-,.L' Pt,-'.,tb'(t
+**4u
e"akr a-"'[z,..-'g"- bate-
a-h'
\>ose&lLs
-is
lsoszbr T'','L tB=AB
ftf3
---be'
&DC-
ln'H'
beza''asc- *l+'
,
2's
btca-st'o Cnv'
bJ trbs$"*u*o^
.->
:
a
Proving the A:ngIe'$isedofTheorefiand tfwBythagorean Thoorerh GSRf,:4 l/utrlrksheet'l Guido 3
Prove166:py1h*6rcafi:Bborem'usingtrlerglesimihtjty.
:'
., ,: i
Given: A right triangle with an altitude (height) drawn from the right angle to the hypotenuse.
Prove:
Lz{y
,
.7
a'+b'=c'
1
I
Q.
r
--i riXh*
C-
A lel+ *.
,Ae^ a
*A
bg AA
bu --: (r
d
^
uLrln'-
A rg [v
G-
+ 6-
d,'.1rj lnol.t,
15 AA
+:ft -7 J*= .o-
AtL
.nz w
a-r
'csJ
cte-:,
l
by
add-'{' "'^-
&?+b-
eA +c
a-+b-
"Cr*qZ FaA6.,YJ$.lli+,
r
1u-5
o b,t"l;
G.SRT.4 WORKSHEET #1B
NAME: _______________________________
1
2. Prove the Angle Bisector Theorem.
The Angle Bisector Theorem states that an angle bisector of an angle of a triangle divides the opposite side in
two segments that are proportional to the other two sides of the triangle.
Given: ∆ABC where BD is an angle bisector.
Prove:
B
AB AD
=
BC DC
o
C
To get this proof moving I’m going to give you a little help – Create the
D
A
auxiliary line AE parallel to BD while also extending side CB until they
meet at point E.
o
E
B
o
o
C
D
A
HINT:
Have you seen this
relationship before?
E
B
C
A
D
G.SRT.4 WORKSHEET #1B
2
3. After figuring out the proof for the Angle Bisector Theorem a Kylie asks; “Could we have done the
auxiliary line parallel line to AB through D instead?” The teacher responds with “Great question Kylie, I’m
not sure but it looks like it should work because we make a ‘similar’ looking diagram”. See if Kylie is correct.
Given: ∆ABC where BD is an angle bisector.
Prove:
B
AB AD
=
BC DC
o
o
C
D
A
B
o
E
o
C
D
A
HINT:
Have you seen this
relationship before?
B
E
C
A
D