Let the measure of the smaller angle be $x$.

Since the angle is complementary, its complement is $90^\circ - x$.

According to the problem, the measure of the larger angle is nine times the measure of the smaller angle, so:

$90^\circ - x = 9x$

Now, solve for $x$:

$90^\circ = 10x$ $x = \frac{90^\circ}{10} = 9^\circ$

Thus, the smaller angle measures $9^\circ$.

To find the larger angle, substitute $x = 9^\circ$ into $90^\circ - x$:

$90^\circ - 9^\circ = 81^\circ$

So, the larger angle measures $81^\circ$.

Final Answer:

• The smaller angle measures $9^\circ$.
• The larger angle measures $81^\circ$.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the sum of two complementary angles?
2. How do you define complementary and supplementary angles?
3. Can an obtuse angle ever be part of a complementary pair?
4. What are complementary angles used for in geometry problems?
5. How would the problem change if the larger angle was eight times the smaller angle?

Tip: In complementary angle problems, always remember that their sum is exactly $90^\circ$.