To calculate the size of the angle $m$, we can use the property of the exterior angles formed by extending the sides of the pentagon. Here's the step-by-step solution:

1. The sum of the exterior angles of any polygon is always $360^\circ$, regardless of the number of sides. In this case, the exterior angles are the angles between the extensions of the sides and the interior angles of the pentagon.

2. From the image, the five interior angles of the pentagon are labeled as: $78^\circ, \ 70^\circ, \ 80^\circ, \ 67^\circ, \text{ and one unknown angle }.$

3. Let's calculate the sum of the interior angles of a pentagon: $\text{Sum of interior angles of a pentagon} = (5-2) \times 180^\circ = 540^\circ.$ We already know four of the interior angles, so: $78^\circ + 70^\circ + 80^\circ + 67^\circ = 295^\circ.$ Therefore, the fifth interior angle is: $540^\circ - 295^\circ = 245^\circ.$

4. The exterior angle corresponding to each interior angle is supplementary (i.e., adds up to $180^\circ$) with the interior angle. So the exterior angle corresponding to the unknown interior angle will be: $180^\circ - 245^\circ = 115^\circ.$