Arc BC is is subtended by the inscribed angle BAC.

Theorem to use:
The measure of an arc subtended by an inscribed angle is twice
the measure of the inscribed angle. 

Since the sum of the measures of the angles of triangle ABC is
180°, 

m(angle A) + m(angle B) + m(angle C) = 180°

              m(angle A) + 68° + 62° = 180°

                   m(angle A) + 130° = 180°
                               -130°  -130°
                 ---------------------------
                          m(angle A) =  50°

So by the theorem, the measure of arc BC is twice 50° or 100°.

Edwin