①如图①，$\because AB=AC$，$BD=CD$，$CD=AD$，
$\therefore \angle B=\angle C=\angle BAD=\angle CAD$，
$\because \angle BAC+\angle B+\angle C=180^{\circ}$，
$\therefore 4\angle B=180^{\circ}$，
$\therefore \angle B=45^{\circ}$，$\angle C=45^{\circ}$，$\angle BAC=90^{\circ}$.
②如图②，$\because AB=AC$，$AD=BD$，$AC=CD$，
$\therefore \angle B=\angle C=\angle BAD$，$\angle CAD=\angle CDA$，
$\because \angle CDA=\angle B+\angle BAD=2\angle B$，
$\therefore \angle BAC=3\angle B$，
$\because \angle BAC+\angle B+\angle C=180^{\circ}$，
$\therefore 5\angle B=180^{\circ}$，
$\therefore \angle B=36^{\circ}$，$\angle C=36^{\circ}$，$\angle BAC=108^{\circ}$.
③如图③，$\because AB=AC$，$AD=BD=BC$，
$\therefore \angle B=\angle C$，$\angle A=\angle ABD$，$\angle BDC=\angle C$，
$\because \angle BDC=\angle A+\angle ABD=2\angle A$，
$\therefore \angle ABC=\angle C=2\angle A$，
$\because \angle A+\angle ABC+\angle C=180^{\circ}$，
$\therefore 5\angle A=180^{\circ}$，
$\therefore \angle A=36^{\circ}$，$\angle C=72^{\circ}$，$\angle ABC=72^{\circ}$.
④如图④，$\because AB=AC$，$AD=BD$，$CD=BC$，
$\therefore \angle ABC=\angle C$，$\angle A=\angle ABD$，$\angle CDB=\angle CBD$，
$\because \angle BDC=\angle A+\angle ABD=2\angle A$，
$\therefore \angle ABC=\angle C=3\angle A$，
$\because \angle A+\angle ABC+\angle C=180^{\circ}$，
$\therefore 7\angle A=180^{\circ}$，
$\therefore \angle A=(\frac{180}{7})^{\circ}$，$\angle C=(\frac{540}{7})^{\circ}$，$\angle ABC=(\frac{540}{7})^{\circ}$.
故答案为：$108°或90°或36°或(\frac{180}{7})°$.

①如图①，$\because AB=AC$，$BD=CD$，$CD=AD$，
$\therefore \angle B=\angle C=\angle BAD=\angle CAD$，
$\because \angle BAC+\angle B+\angle C=180^{\circ}$，
$\therefore 4\angle B=180^{\circ}$，
$\therefore \angle B=45^{\circ}$，$\angle C=45^{\circ}$，$\angle BAC=90^{\circ}$.
②如图②，$\because AB=AC$，$AD=BD$，$AC=CD$，
$\therefore \angle B=\angle C=\angle BAD$，$\angle CAD=\angle CDA$，
$\because \angle CDA=\angle B+\angle BAD=2\angle B$，
$\therefore \angle BAC=3\angle B$，
$\because \angle BAC+\angle B+\angle C=180^{\circ}$，
$\therefore 5\angle B=180^{\circ}$，
$\therefore \angle B=36^{\circ}$，$\angle C=36^{\circ}$，$\angle BAC=108^{\circ}$.
③如图③，$\because AB=AC$，$AD=BD=BC$，
$\therefore \angle B=\angle C$，$\angle A=\angle ABD$，$\angle BDC=\angle C$，
$\because \angle BDC=\angle A+\angle ABD=2\angle A$，
$\therefore \angle ABC=\angle C=2\angle A$，
$\because \angle A+\angle ABC+\angle C=180^{\circ}$，
$\therefore 5\angle A=180^{\circ}$，
$\therefore \angle A=36^{\circ}$，$\angle C=72^{\circ}$，$\angle ABC=72^{\circ}$.
④如图④，$\because AB=AC$，$AD=BD$，$CD=BC$，
$\therefore \angle ABC=\angle C$，$\angle A=\angle ABD$，$\angle CDB=\angle CBD$，
$\because \angle BDC=\angle A+\angle ABD=2\angle A$，
$\therefore \angle ABC=\angle C=3\angle A$，
$\because \angle A+\angle ABC+\angle C=180^{\circ}$，
$\therefore 7\angle A=180^{\circ}$，
$\therefore \angle A=(\frac{180}{7})^{\circ}$，$\angle C=(\frac{540}{7})^{\circ}$，$\angle ABC=(\frac{540}{7})^{\circ}$.
故答案为：$108°或90°或36°或(\frac{180}{7})°$.