1. **State the problem:** We need to find the measure of angle $g$ in a figure composed of a regular octagon, a regular hexagon, a square, and an equilateral triangle. 2. **Recall the interior angles of regular polygons:** - Square: each interior angle is $90^\circ$ - Regular hexagon: each interior angle is $120^\circ$ - Regular octagon: each interior angle is $135^\circ$ - Equilateral triangle: each interior angle is $60^\circ$ 3. **Given angles:** - $c = 120$ - $d = 42$ - $e = 135$ - $f = 15$ 4. **Analyze angle $g$:** Angle $g$ is adjacent to angle $d$ (42) and angle $f$ (15) around a point on the figure. 5. **Sum of angles around a point:** The sum of angles around a point is $360^\circ$. 6. **Calculate angle $g$:** $$g = 360 - (d + f + e) = 360 - (42 + 15 + 135)$$ $$g = 360 - 192 = 168$$ 7. **Final answer:** The measure of angle $g$ is $168$ degrees.