jmc

The angles numbered 1, 3, and 5 are the three interior angles of a triangle, so they sum to $180^{\circ }$. Similarly, the angles numbered 2, 4, and 6 are the three interior angles of a triangle, so they also sum to $180^{\circ }$. Combining these, the sum of all six angle measures is $180^{\circ }+180^{\circ }=\boxed{360^{\circ }}$.