1. **State the problem:** We need to find the size of angle $m$ in a figure where six rays extend from a central point, creating six angles around that point. 2. **Recall the rule:** The sum of all angles around a point is always $360^\circ$. 3. **Given angles:** $13^\circ$, $24^\circ$, and $15^\circ$ are three of the six angles. 4. **Calculate the sum of the given angles:** $$13^\circ + 24^\circ + 15^\circ = 52^\circ$$ 5. **Since the six angles around the point sum to $360^\circ$, the sum of the other three angles (including $m$) is:** $$360^\circ - 52^\circ = 308^\circ$$ 6. **Assuming the other three angles are equal or that $m$ is the only unknown angle adjacent to the given ones, and the problem implies $m$ is the remaining angle adjacent to these three, then $m$ is:** $$m = 360^\circ - (13^\circ + 24^\circ + 15^\circ) = 308^\circ$$ 7. **Therefore, the size of angle $m$ is $308^\circ$.**