Understanding the Problem: Angles and Complements

The question asks us to find the value of the smaller of two angles, given a relationship between one angle and its complement. Let's first understand what complementary angles are.

Complementary angles are two angles that add up to exactly 90 degrees (\(90^\circ\)). If we have an angle, its complement is the angle that, when added to the first angle, results in \(90^\circ\).

Let the angle we are trying to find be represented by \(x\). The question states that this angle is related to its complement. If the angle is \(x\), its complement must be \(90^\circ - x\), because \(x + (90^\circ - x) = 90^\circ\).

The problem gives us a specific relationship:

• An angle (\(x\)) is 60° more than one-fifth of its complement (\(90^\circ - x\)).

We can translate this statement into a mathematical equation.

Setting up the Equation

Let the angle be \(x\).

Its complement is \(90^\circ - x\).

One-fifth of its complement is \(\frac{1}{5} \times (90^\circ - x)\).

The angle \(x\) is 60° more than this value. So, the equation is:

\(x = \frac{1}{5}(90^\circ - x) + 60^\circ\)

Solving the Equation for the Angle

Now, we need to solve this linear equation for \(x\). We can start by multiplying the entire equation by 5 to get rid of the fraction:

\(5 \times x = 5 \times \left( \frac{1}{5}(90^\circ - x) + 60^\circ \right)\)

\(5x = 5 \times \frac{1}{5}(90^\circ - x) + 5 \times 60^\circ\)

\(5x = (90^\circ - x) + 300^\circ\)

Now, remove the parentheses and combine the constant terms on the right side:

\(5x = 90^\circ - x + 300^\circ\)

\(5x = 390^\circ - x\)

Next, bring the term with \(x\) from the right side to the left side by adding \(x\) to both sides:

\(5x + x = 390^\circ\)

\(6x = 390^\circ\)

Finally, divide both sides by 6 to find the value of \(x\):

\(x = \frac{390^\circ}{6}\)

\(x = 65^\circ\)

So, the angle we defined as \(x\) is \(65^\circ\).

Finding the Smaller Angle

The question asks for the smaller angle in degrees. We have found one angle, which is \(65^\circ\).

The complement of this angle is \(90^\circ - x\). Let's calculate it:

Complement = \(90^\circ - 65^\circ = 25^\circ\)

The two angles that are complementary and satisfy the condition are \(65^\circ\) and \(25^\circ\).

Comparing these two angles, \(25^\circ\) is smaller than \(65^\circ\).

Therefore, the smaller angle is \(25^\circ\).

Verifying the Solution

Let's check if our angles \(65^\circ\) and \(25^\circ\) satisfy the original condition:

• The angle is \(65^\circ\).
• Its complement is \(25^\circ\).
• One-fifth of its complement is \(\frac{1}{5} \times 25^\circ = 5^\circ\).
• Is the angle \(65^\circ\) 60° more than 5°? \(5^\circ + 60^\circ = 65^\circ\). Yes, it is.

The condition is satisfied. The two complementary angles are \(65^\circ\) and \(25^\circ\). The smaller angle is \(25^\circ\).

Conclusion

Based on our calculations, the smaller angle is \(25^\circ\).

Let's look at the given options:

Our calculated smaller angle, \(25^\circ\), matches Option C.

Revision Table: Key Concepts

Additional Information: More on Angles

Besides complementary angles, there are other important types of angle relationships:

• Supplementary Angles: Two angles that add up to 180 degrees (\(180^\circ\)). If an angle is \(y\), its supplement is \(180^\circ - y\).
• Adjacent Angles: Two angles that share a common vertex and a common side, but have no common interior points.
• Vertical Angles: Two non-adjacent angles formed by two intersecting lines. Vertical angles are always equal.
• Angles on a Straight Line: Angles that form a straight line add up to \(180^\circ\) (they are supplementary).
• Angles Around a Point: Angles that meet at a single point and complete a full circle add up to \(360^\circ\).

Understanding these angle relationships is fundamental in geometry problems.