Complementary Angles Definition

Two angles are complementary if the sum of their measures is exactly $90^\circ$.

Setting Up the Equation

Let the measure of one angle be $x$. Since the angles are complementary, the measure of the other angle is $90^\circ - x$.

The problem states that the measure of an angle is three times the measure of its complement. We can write this relationship as an equation:

$x = 3 \times (90^\circ - x)$

Solving for the Angle Measure

1. Distribute the 3 on the right side of the equation:
   $x = 270^\circ - 3x$
2. Add $3x$ to both sides to gather the $x$ terms:
   $x + 3x = 270^\circ$
   $4x = 270^\circ$
3. Divide by 4 to solve for $x$:
   $x = \frac{270^\circ}{4}$
   $x = 67.5^\circ$

Finding Both Angles

We found one angle to be $x = 67.5^\circ$.

The measure of its complement is $90^\circ - x = 90^\circ - 67.5^\circ = 22.5^\circ$.

So, the two angles are $22.5^\circ$ and $67.5^\circ$.

We can check: Is $67.5^\circ$ three times $22.5^\circ$? Yes, $3 \times 22.5^\circ = 67.5^\circ$.

This matches option B.