1. **State the problem:** We need to find the value of $a$ given the angles in a triangle where one angle is $90^\circ$, another is $21^\circ$, and the third is $(2a + 6)^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. 3. **Set up the equation:** Using the given angles, $$90 + 21 + (2a + 6) = 180$$ 4. **Simplify the equation:** $$111 + 2a + 6 = 180$$ $$2a + 117 = 180$$ 5. **Solve for $a$:** $$2a = 180 - 117$$ $$2a = 63$$ $$a = \frac{63}{2} = 31.5$$ 6. **Conclusion:** The value of $a$ is $31.5$.