Question
Prove that:
(i) $\triangle A B C \cong \triangle A D C$
(ii) $\angle B=\angle D$
(iii) $A C$ bisects angle $D C B$
(i) $\triangle A B C \cong \triangle A D C$
(ii) $\angle B=\angle D$
(iii) $A C$ bisects angle $D C B$
✓
Answer
Given: In the figure,
$
A B=A D, C B=C D
$
To prove: $\triangle A B C \cong \triangle A D C$
$
\angle B=\angle D
$
AC bisects angle DCB
Proof: In $\triangle A B C$ and $\triangle A D C$,
$A C=A C$ ..............(common)
$A B=A D$ ..............(given)
$C B=C D$ ..............(given)
(i) $\therefore \triangle ABC \cong \triangle ADC$ .................(SSS aciom)
(ii) $\therefore \angle B=D$ .................(c.p.c.t.)
$
\angle B C A=\angle D C A
$
(iii) $\therefore A C$ bisects $\angle D C B$
$
A B=A D, C B=C D
$
To prove: $\triangle A B C \cong \triangle A D C$
$
\angle B=\angle D
$
AC bisects angle DCB
Proof: In $\triangle A B C$ and $\triangle A D C$,
$A C=A C$ ..............(common)
$A B=A D$ ..............(given)
$C B=C D$ ..............(given)
(i) $\therefore \triangle ABC \cong \triangle ADC$ .................(SSS aciom)
(ii) $\therefore \angle B=D$ .................(c.p.c.t.)
$
\angle B C A=\angle D C A
$
(iii) $\therefore A C$ bisects $\angle D C B$
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