2013-I-07

$\begin{array}{ll}
BE=CE & \text{(Given)} \\
\angle EBC = \angle ECB & \text{(base $\angle$s, isos. $\Delta$)} \\
\end{array}$

In $\Delta ABC$ and $\Delta DCB$,

$\begin{array}{ll}
BC=CB & \text{(Common side)} \\
\angle BAC = \angle CDB & \text{(Given)} \\
\angle ACB = \angle DBC & \text{(Proved)}
\end{array}$

$\therefore \Delta ABC \cong \Delta DCB \text{ (AAS)}$.

1. There are 3 pairs of congruent triangles. They are $\Delta ABC \cong \Delta DCB$, $\Delta ABE \cong \Delta DCE$ and $\Delta ABD \cong \Delta DCA$.
2. There are 4 pairs of similar triangles. They are $\Delta ABC \sim \Delta DCB$, $\Delta ABE \sim \Delta DCE$, $\Delta ABD \sim \Delta DCA$ and $\Delta AED \sim \Delta BEC$.