1. **Problem statement:** We need to find the size of angle $g$ in a figure where a triangle shares a side with an irregular pentagon. Given angles are 68°, 77°, 116°, 101°, and 21°. 2. **Key fact:** The sum of interior angles of a triangle is always $180^\circ$. 3. **Identify the triangle angles:** The triangle has angles 77°, $g$, and the angle adjacent to 68° (which is supplementary to 68° because they form a straight line). So, that angle is $180^\circ - 68^\circ = 112^\circ$. 4. **Apply the triangle angle sum rule:** $$ 77^\circ + g + 112^\circ = 180^\circ $$ 5. **Solve for $g$:** $$ g = 180^\circ - 77^\circ - 112^\circ = 180^\circ - 189^\circ = -9^\circ $$ This negative value indicates a misinterpretation of the angles. Let's reconsider the triangle angles carefully. 6. **Re-examining the triangle:** The triangle shares a side with the pentagon. The angles inside the triangle are $g$, 77°, and the angle adjacent to 21° (since 21° is inside the pentagon near the shared side). The angle adjacent to 21° is $180^\circ - 21^\circ = 159^\circ$. 7. **Sum of triangle angles:** $$ 77^\circ + g + 159^\circ = 180^\circ $$ 8. **Solve for $g$:** $$ g = 180^\circ - 77^\circ - 159^\circ = 180^\circ - 236^\circ = -56^\circ $$ Again negative, so this is not correct. Let's try another approach. 9. **Use the pentagon angle sum:** The sum of interior angles of a pentagon is $$ (5-2) \times 180^\circ = 540^\circ $$ 10. **Sum the known pentagon angles:** $$ 68^\circ + 116^\circ + 101^\circ + 21^\circ + x = 540^\circ $$ where $x$ is the pentagon angle adjacent to $g$. 11. **Calculate $x$:** $$ x = 540^\circ - (68^\circ + 116^\circ + 101^\circ + 21^\circ) = 540^\circ - 306^\circ = 234^\circ $$ 12. **Since $g$ and $x$ are adjacent angles on a straight line, they are supplementary:** $$ g + x = 180^\circ $$ 13. **Solve for $g$:** $$ g = 180^\circ - 234^\circ = -54^\circ $$ Negative again, so $x$ cannot be 234°. This suggests the pentagon is irregular and the angle adjacent to $g$ is not $x$ but rather the 21° angle. 14. **Check if $g$ and 21° are supplementary:** $$ g + 21^\circ = 180^\circ \Rightarrow g = 159^\circ $$ 15. **Check triangle sum with $g=159^\circ$ and 77°:** $$ 159^\circ + 77^\circ + ? = 180^\circ $$ The third angle would be $$ 180^\circ - 159^\circ - 77^\circ = -56^\circ $$ which is impossible. 16. **Conclusion:** The only consistent approach is to consider the triangle angles as 68°, 77°, and $g$. 17. **Sum of triangle angles:** $$ 68^\circ + 77^\circ + g = 180^\circ $$ 18. **Solve for $g$:** $$ g = 180^\circ - 68^\circ - 77^\circ = 35^\circ $$ **Final answer:** $$ g = 35^\circ $$