Let's break down the problem step by step:

1. Let the angle be $x$.

2. The supplement of an angle is given by $180^\circ - x$, and the complement of the angle is $90^\circ - x$.

3. According to the problem, the measure of the supplement exceeds 4 times the measure of the complement by 9. So, we set up the equation:

$180^\circ - x = 4(90^\circ - x) + 9$

4. Now, let's solve this equation:

$180 - x = 4(90 - x) + 9$ $180 - x = 360 - 4x + 9$ $180 - x = 369 - 4x$

5. Move the terms involving $x$ to one side and the constants to the other side:

$- x + 4x = 369 - 180$ $3x = 189$

6. Solve for $x$:

$x = \frac{189}{3} = 63$

7. The angle $x$ is $63^\circ$. Now, the complement of the angle is:

$90^\circ - x = 90^\circ - 63^\circ = 27^\circ$

Thus, the measure of the complement is $27^\circ$.

Would you like further details on the solution?

Here are some related questions:

1. What is the relationship between supplementary and complementary angles?
2. Can you provide another example of a problem involving supplementary and complementary angles?
3. What happens if an angle exceeds its complement by a certain value?
4. How can you solve an equation involving multiple angles using algebra?
5. What real-world scenarios might involve supplementary and complementary angles?

Tip: Always remember, supplementary angles add up to $180^\circ$, and complementary angles add up to $90^\circ$.