Angle Relationship: Understanding Supplement and Complement

This problem involves understanding the relationship between an angle and its supplementary and complementary angles. Let's break down the concepts:

• Complementary Angles: Two angles are complementary if their sum is exactly 90 degrees ($90^\circ$). If one angle is $x$, its complement is $(90^\circ - x)$.
• Supplementary Angles: Two angles are supplementary if their sum is exactly 180 degrees ($180^\circ$). If one angle is $x$, its supplement is $(180^\circ - x)$.

Angle Calculation: Setting Up the Equation

The question states that the measure of the supplement of an angle is four times the measure of its complement. Let the measure of the angle be represented by the variable $x$.

• The measure of the complement of the angle is: $90^\circ - x$.
• The measure of the supplement of the angle is: $180^\circ - x$.

According to the problem statement, we can write the equation:

Measure of Supplement = 4 × Measure of Complement

Using the expressions above, the equation becomes:

$$180^\circ - x = 4(90^\circ - x)$$

Solving for the Angle Measure

Now, we need to solve this algebraic equation for $x$. Let's follow the steps:

1. Distribute the 4 on the right side of the equation:

   $$180^\circ - x = 4 \times 90^\circ - 4 \times x$$

   $$180^\circ - x = 360^\circ - 4x$$

2. To solve for $x$, we want to get all the terms with $x$ on one side and the constant terms on the other side. Add $4x$ to both sides:

   $$180^\circ - x + 4x = 360^\circ - 4x + 4x$$

   $$180^\circ + 3x = 360^\circ$$

3. Subtract $180^\circ$ from both sides:

   $$180^\circ + 3x - 180^\circ = 360^\circ - 180^\circ$$

   $$3x = 180^\circ$$

4. Finally, divide by 3 to find the value of $x$:

   $$x = \frac{180^\circ}{3}$$

   $$x = 60^\circ$$

Verification of the Angle

Let's check if the angle $x = 60^\circ$ satisfies the condition given in the question.

• The angle is $60^\circ$.
• Its complement is $90^\circ - 60^\circ = 30^\circ$.
• Its supplement is $180^\circ - 60^\circ = 120^\circ$.

Now, let's see if the supplement ($120^\circ$) is four times the complement ($30^\circ$):

$$4 \times 30^\circ = 120^\circ$$

Since $120^\circ = 120^\circ$, our calculated value of $x = 60^\circ$ is correct.

Final Answer

The measure of the angle is $60^\circ$.