Solving Complementary Angles Algebraically

Two angles are complementary if their sum is equal to 90 degrees. Let the measures of the two angles be '$x$' and '$y$'.

From the problem statement, we have the following conditions:

• The angles are complementary: $x + y = 90^\circ$.
• One angle's measure is 10 degrees more than three times the other: Let $x = 3y + 10^\circ$.

Step-by-Step Solution

1. Substitute the expression for '$x$' from the second condition into the first equation:

   $(3y + 10^\circ) + y = 90^\circ$

2. Combine like terms:

   $4y + 10^\circ = 90^\circ$

3. Isolate the term with '$y$' by subtracting $10^\circ$ from both sides:

   $4y = 90^\circ - 10^\circ$

   $4y = 80^\circ$

4. Solve for '$y$' by dividing both sides by 4:

   $y = \frac{80^\circ}{4}$

   $y = 20^\circ$

5. Find the measure of the other angle, '$x$':

   $x = 3y + 10^\circ = 3(20^\circ) + 10^\circ = 60^\circ + 10^\circ = 70^\circ$

6. Identify the smaller angle. The two angles are $20^\circ$ and $70^\circ$. The smaller angle is $20^\circ$.

Therefore, the measure of the smaller angle is $20^\circ$.