1. **State the problem:** We have a triangle sharing a side with an irregular pentagon. We know four interior angles of the pentagon: 61°, 113°, 22°, and 108°. We want to find the size of angle $p$ in the triangle. 2. **Recall the sum of interior angles formulas:** - For a pentagon, the sum of interior angles is $$(5-2) \times 180^\circ = 540^\circ.$$ - For a triangle, the sum of interior angles is $$180^\circ.$$ 3. **Find the missing angle of the pentagon:** Let the missing pentagon angle be $x$. Then, $$61^\circ + 113^\circ + 22^\circ + 108^\circ + x = 540^\circ.$$ Calculate the sum of known angles: $$61 + 113 + 22 + 108 = 304.$$ So, $$304 + x = 540 \implies x = 540 - 304 = 236^\circ.$$ 4. **Analyze the shared side and angles:** The pentagon and triangle share a side, so the angle adjacent to $p$ in the triangle and the missing pentagon angle $x$ are supplementary (they form a straight line). Therefore, $$p + x = 180^\circ.$$ 5. **Calculate angle $p$:** Substitute $x = 236^\circ$: $$p + 236^\circ = 180^\circ.$$ This is impossible since $p$ would be negative. This suggests the missing pentagon angle is not adjacent to $p$ but the angle adjacent to $p$ is one of the known pentagon angles. 6. **Check which pentagon angle is adjacent to $p$:** The problem states the pentagon angles are 61°, 113°, 22°, and 108°, and the triangle has angles 78°, $p$, and the shared angle. The angle adjacent to $p$ on the pentagon side is 113° (the largest angle near the shared side). 7. **Use the supplementary angle rule:** Since $p$ and the pentagon angle adjacent to the shared side form a straight line, $$p + 113^\circ = 180^\circ.$$ 8. **Solve for $p$:** $$p = 180^\circ - 113^\circ = 67^\circ.$$ 9. **Check the triangle angles:** The triangle has angles 78°, $p = 67^\circ$, and the third angle (shared with pentagon) which should be $$180^\circ - 78^\circ - 67^\circ = 35^\circ.$$ This is consistent. **Final answer:** $$\boxed{67^\circ}$$