解：（1）$\because \angle ABC = 120^{\circ}，$
$\therefore \angle ABA_{1}=\angle CBC_{1}=180^{\circ}-\angle ABC = 180^{\circ}-120^{\circ}=60^{\circ}，$
$\therefore$旋转角的度数为$60^{\circ}。$
（2）由题意，得$\triangle ABC\cong\triangle A_{1}BC_{1}，$
$\therefore AB = A_{1}B，$$\angle C=\angle C_{1}。$
由（1），知$\angle ABA_{1}=60^{\circ}，$
$\therefore \triangle A_{1}AB$是等边三角形，
$\therefore \angle BAA_{1}=60^{\circ}，$
$\therefore \angle BAA_{1}=\angle CBC_{1}，$
$\therefore AA_{1}// BC，$
$\therefore \angle A_{1}AC=\angle C，$
$\therefore \angle A_{1}AC=\angle C_{1}。$
