1. **State the problem:** We need to find the size of angle $q$ in a composite figure consisting of a triangle and an irregular pentagon sharing a side. 2. **Identify known angles:** The triangle has angles $q$, $79^\circ$, and $63^\circ$. The pentagon has angles $103^\circ$, $27^\circ$, and $114^\circ$. 3. **Use the triangle angle sum rule:** The sum of angles in any triangle is $180^\circ$. 4. **Set up the equation for the triangle:** $$q + 79^\circ + 63^\circ = 180^\circ$$ 5. **Calculate $q$:** $$q = 180^\circ - 79^\circ - 63^\circ$$ $$q = 180^\circ - 142^\circ$$ $$q = 38^\circ$$ 6. **Answer:** The size of angle $q$ is $38^\circ$. This uses the fundamental property that the sum of interior angles in a triangle is always $180^\circ$, allowing us to find the unknown angle by subtracting the known angles from $180^\circ$.