Let the two complementary angles be $x$ and $y$. By definition, the sum of complementary angles is $90^\circ$. We also know that one angle is five times the other. This gives us the following equations:

$x + y = 90$ $y = 5x$

Step 1: Substitute $y = 5x$ into $x + y = 90$:

$x + 5x = 90$

Step 2: Simplify and solve for $x$:

$6x = 90 \implies x = \frac{90}{6} = 15$

Step 3: Find $y$ using $y = 5x$:

$y = 5(15) = 75$

Final Answer:

The two complementary angles are: $x = 15^\circ \quad \text{and} \quad y = 75^\circ$

Would you like more explanation or help with related concepts? Here are some follow-up questions to deepen your understanding:

1. How do supplementary angles differ from complementary angles?
2. What happens if the ratio between the two angles is changed?
3. Can two complementary angles both be greater than $45^\circ$? Why or why not?
4. How can we use algebra to solve problems involving other relationships between angles?
5. What real-life applications involve complementary angles?

Tip: Always verify your solution by checking that the angles add up to $90^\circ$.