The measure of an angle for which the measure of the supplement is four times the measure of the complement is

Question

Accepted Answer

To determine why Option 3 (60°) is the correct answer, let's analyze the problem step by step.

### Explanation for Option 3 (60°):

1. **Definitions**:
   - The **supplement** of an angle is what, when added to the angle, equals 180°. So, if the angle is $ x $, the supplement is $ 180° - x $.
   - The **complement** of an angle is what, when added to the angle, equals 90°. So, if the angle is $ x $, the complement is $ 90° - x $.

2. **Given Condition**:
   - The measure of the supplement is four times the measure of the complement.
   - Mathematically, this can be expressed as:
     $$
     180° - x = 4 \times (90° - x)
     $$

3. **Solving the Equation**:
   - Start with the equation:
     $$
     180° - x = 4 \times (90° - x)
     $$
   - Expand the right side:
     $$
     180° - x = 360° - 4x
     $$
   - Rearrange the equation to solve for $ x $:
     $$
     3x = 180°
     $$
   - Divide both sides by 3:
     $$
     x = 60°
     $$

Thus, the angle that satisfies the condition is 60°, confirming that Option 3 is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (30°)**:
   - If $ x = 30° $, then the supplement is $ 180° - 30° = 150° $ and the complement is $ 90° - 30° = 60° $.
   - The condition requires the supplement to be four times the complement: $ 4 \times 60° = 240° $, which does not equal 150°.
   - Therefore, 30° does not satisfy the condition.

2. **Option 2 (45°)**:
   - If $ x = 45° $, then the supplement is $ 180° - 45° = 135° $ and the complement is $ 90° - 45° = 45° $.
   - The condition requires the supplement to be four times the complement: $ 4 \times 45° = 180° $, which does not equal 135°.
   - Therefore, 45° does not satisfy the condition.

3. **Option 4 (75°)**:
   - If $ x = 75° $, then the supplement is $ 180° - 75° = 105° $ and the complement is $ 90° - 75° = 15° $.
   - The condition requires the supplement to be four times the complement: $ 4 \times 15° = 60° $, which does not equal 105°.
   - Therefore, 75° does not satisfy the condition.

In conclusion, only Option 3 (60°) satisfies the given condition, making it the correct answer.