Question

In given figure if AC = BC, $\angle$DCA = $\angle$ECB and $\angle$DBC = $\angle$EAC, then DC = __.

Answer 1

We have,
$\angle$DCA = $\angle$ECB
$\Rightarrow$ $\angle$DCA + $\angle$ECD = $\angle$ECB + $\angle$ECD
$\Rightarrow$ $\angle$ECA = $\angle$DCB .......(i)

Now, in $\Delta$s DBC and EAC, we have
$\angle$DCB = $\angle$ECA [From (i)]
BC = AC [Given]
and $\angle$DBC = $\angle$EAC [Given]

So, by ASA criterion of congruence
$\Delta$DBC $\cong$ $\Delta$EAS
$\Rightarrow$ DC = EC [ Correcponding parts of congruent triangles are equal ]