1. **State the problem:** We are given angles in a triangle and need to find the value of $a$. 2. **Given angles:** $(2a + 6)^\circ$, $21^\circ$, $90^\circ$, $90^\circ$, and $a^\circ$. 3. **Key fact:** The sum of angles on a straight line is $180^\circ$ (supplementary angles). 4. **Set up the equation:** Since $(2a + 6)^\circ$ and $21^\circ$ are supplementary, $$ (2a + 6) + 21 = 180 $$ 5. **Simplify:** $$ 2a + 6 + 21 = 180 $$ $$ 2a + 27 = 180 $$ 6. **Solve for $a$:** $$ 2a = 180 - 27 $$ $$ 2a = 153 $$ $$ a = \frac{153}{2} = 76.5 $$ 7. **Check options:** None of the options (28, 62, 69, 83) match $76.5$ exactly, so re-examine the problem. 8. **Re-examining the problem:** The user wrote $2a + 6 = 21$ in their working, which suggests the angles $(2a + 6)^\circ$ and $21^\circ$ are equal or related differently. 9. **If $(2a + 6)^\circ = 21^\circ$, then:** $$ 2a + 6 = 21 $$ $$ 2a = 15 $$ $$ a = 7.5 $$ 10. **This $a=7.5$ is not among the options either.** 11. **Considering the triangle angles:** The triangle has angles $a^\circ$, $21^\circ$, and $90^\circ$. Sum of triangle angles is $180^\circ$: $$ a + 21 + 90 = 180 $$ $$ a + 111 = 180 $$ $$ a = 69 $$ 12. **Answer:** $a = 69$ which matches option C. --- **Final answer:** $\boxed{69}$