ABCD IS A QUADRILATERAL WHERE AB =AD AND ANGLE B +ANGLE D =ANGLE C . PROVE THAT AC=AB?
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Given A quadrilateral ABCD in which AB=AD and the bisectors of ∠BAC and ∠CAD meet the sides BC and CD at E and F respectively.
To prove EF||BD
Construction Join AC, BD and EF.
Proof In △CAB, AE is the bisector of ∠BAC.
∴
AB
AC
=
BE
CE
.......(i)
In △ACD, AF is the bisector of ∠CAD.
∴
AD
AC
=
DF
CF
⇒
AB
AC
=
DF
CF
[∵ AD=AB]........(ii)
From (i) and (ii), we get
BE
CE
=
DF
CF
⇒
EB
CE
=
FD
CF
Thus, in △CBD, E and F divide the sides CB and CD respectively in the same ratio. Therefore, by the converse of Thale's Theorem, we have
EF∣∣BD
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