1. **Stating the problem:** We are given a geometric figure with angles labeled as $A = (3b - 5)^\circ$, $B = 88^\circ$, $C = (c + 5)^\circ$, $D = e^\circ$, $R = 72^\circ$, $S = 5d^\circ$, $P = 115^\circ$, and $Q = 2a^\circ$. We need to find the values of $a$, $b$, $c$, $d$, and $e$. 2. **Understanding the problem:** Since the figure is not explicitly described, we assume these angles are related by geometric rules such as angles on a straight line sum to $180^\circ$, angles around a point sum to $360^\circ$, or angles in triangles sum to $180^\circ$. 3. **Using angle sum rules:** - From the options, $a$, $b$, $c$, $d$, and $e$ are integers. - We use the given angle values and expressions to set up equations. 4. **From angle $Q = 2a^\circ$ and $P = 115^\circ$:** Assuming $P$ and $Q$ are angles on a straight line, $$ 2a + 115 = 180 $$ $$ 2a = 180 - 115 = 65 $$ $$ a = \frac{65}{2} = 32.5 $$ This does not match any option, so maybe $P$ and $Q$ are not supplementary. 5. **From angle $S = 5d^\circ$ and $R = 72^\circ$:** Assuming $R$ and $S$ are angles in a triangle with $P$ or $Q$ or other angles, or supplementary. 6. **Check option A:** $a=44$, $b=40$, $c=67$, $d=17$, $e=88$ - Calculate $Q = 2a = 2 \times 44 = 88^\circ$ - Calculate $A = 3b - 5 = 3 \times 40 - 5 = 120 - 5 = 115^\circ$ - Calculate $C = c + 5 = 67 + 5 = 72^\circ$ - Calculate $S = 5d = 5 \times 17 = 85^\circ$ - $D = e = 88^\circ$ 7. **Check if angles sum correctly:** - For example, if $A$, $B$, and $C$ are angles in a triangle: $$ A + B + C = 115 + 88 + 72 = 275^\circ $$ Not possible for a triangle. 8. **Check option C:** $a=44$, $b=40$, $c=67$, $d=17$, $e=85$ - $Q = 2a = 88^\circ$ - $A = 115^\circ$ - $C = 72^\circ$ - $S = 85^\circ$ - $D = 85^\circ$ 9. **Sum $A + B + C = 115 + 88 + 72 = 275^\circ$ again, no triangle. 10. **Check option B:** $a=44$, $b=40$, $c=77$, $d=17$, $e=85$ - $Q = 88^\circ$ - $A = 115^\circ$ - $C = 82^\circ$ - Sum $A + B + C = 115 + 88 + 82 = 285^\circ$ 11. **Check option D:** $a=44$, $b=35$, $c=67$, $d=44$, $e=88$ - $Q = 88^\circ$ - $A = 3b - 5 = 3 \times 35 - 5 = 105 - 5 = 100^\circ$ - $C = 72^\circ$ - Sum $A + B + C = 100 + 88 + 72 = 260^\circ$ 12. **Conclusion:** The only option where $A = 115^\circ$ matches the given $P = 115^\circ$ is option A or C. 13. **Final answer:** Based on the given data and options, the best match is option A: $a=44$, $b=40$, $c=67$, $d=17$, and $e=88$. **Answer:** $\boxed{a=44, b=40, c=67, d=17, e=88}$