Question

recall that two angles are complementary if the sum of their measures is $90^{circ}$. find the measures of two complementary angles if one angle is $15^{circ}$ more than four times the other angle. the smaller angle measures $\square^{circ}$

Explanation:

Step1: Define variables

Let the smaller angle be \( x \) degrees. Then the larger angle is \( 4x + 15 \) degrees (since it is \( 15^\circ \) more than four times the smaller angle).

Step2: Use complementary angle property

Since the two angles are complementary, their sum is \( 90^\circ \). So we set up the equation:
\( x + (4x + 15) = 90 \)

Step3: Solve the equation

Combine like terms:
\( 5x + 15 = 90 \)

Subtract 15 from both sides:
\( 5x = 90 - 15 \)
\( 5x = 75 \)

Divide both sides by 5:
\( x = \frac{75}{5} \)
\( x = 15 \)

Answer:

The smaller angle measures \( \boldsymbol{15}^\circ \)