Angles of an Isosceles Triangle

Let $ABC$ be a triangle with $\angle A = x^\circ$ and $\angle B = 68 ^\circ$ .

We have the following cases:

Case 1: $\angle A = \angle B$ . Then $\angle C = 180^\circ - 2 \times \angle B = 44 ^\circ$ . Thus $x^\circ = 68 ^\circ$ .

Case 2: $\angle A = \angle C$ . Thus $180^\circ = \angle A + \angle B + \angle C = x^\circ + 68 ^\circ + x^\circ =2x^\circ + 68 ^\circ$ $\Rightarrow x^\circ = 56 ^\circ$ .

Case 3: $\angle B = \angle C$ . Thus $180^\circ = \angle A + \angle B + \angle C = x^\circ + 68 ^\circ + 68 ^\circ =x^\circ + 136 ^\circ$ $\Rightarrow x^\circ = 44 ^\circ$ .

Hence, the sum of the possible values of $x$ is: $68 + 56 + 44 = 168$ .