The Inscribed Angle Theorem states that the measure of an inscribed angle of a circle is half the measure of the central angle that subtends the same arc of the circle. Thus, the measure of the central angle that subtends the same arc has measure $2 \cdot 77^\circ = 154^\circ$.
The length of the arc is therefore $$\frac{154^\circ}{360^\circ} \cdot 2\pi r$$