An angle is equal to two-thirds of its supplementary angle. What is the measure of this angle in degrees?

Introduction / Context:
This is a simple algebraic problem about supplementary angles. Two angles are supplementary if their sum is 180 degrees. The question tells you that one angle is two-thirds of its supplementary angle, and you must find the measure of that angle. Such problems help reinforce the relationship between angles that form a straight line and your ability to set up equations from word statements.

Given Data / Assumptions:

• Let the required angle be x degrees.
• Its supplementary angle is 180° - x.
• The given relationship is: x is two-thirds of its supplementary angle.
• We must find the value of x in degrees.

Concept / Approach:
Use the definition of supplementary angles and translate the phrase "two-thirds of its supplementary angle" into an equation. The basic idea is:
x = (2 / 3) * (180° - x). Then solve this linear equation in x. This approach is very standard for problems involving fractions of angles and their complements or supplements.

Step-by-Step Solution:
Step 1: Let the required angle be x degrees. Step 2: Its supplementary angle is 180° - x. Step 3: The condition given is: x = (2 / 3) * (180° - x). Step 4: Multiply both sides by 3 to clear the fraction: 3x = 2 * (180° - x). Step 5: Expand the right side: 3x = 360° - 2x. Step 6: Add 2x to both sides: 5x = 360°. Step 7: Divide both sides by 5: x = 360° / 5 = 72°. Step 8: Therefore, the angle measures 72 degrees.

Verification / Alternative check:
Compute the supplementary angle: 180° - 72° = 108°. Now check the relationship: two-thirds of 108° is (2 / 3) * 108° = 72°, which matches x. This quick check confirms that our algebraic equation accurately captured the given condition and that the arithmetic steps were performed correctly.

Why Other Options Are Wrong:
If x = 108°, then its supplementary angle is 72°, and x would be greater than its supplement, not two-thirds of it. If x = 54°, then its supplement is 126°, and (2 / 3) * 126° = 84°, which is not 54°. For x = 36°, the supplement is 144°, and (2 / 3) * 144° = 96°, not 36°. An angle of 90° would have a supplement of 90°, and two-thirds of 90° is 60°, not 90°. Only 72° satisfies the given relationship exactly.

Common Pitfalls:
One common error is reversing the relationship and writing 180° - x = (2 / 3)x instead of x = (2 / 3)(180° - x), which leads to a different equation and the wrong answer. Another mistake is adding the angles to get 90° instead of 180°, confusing supplementary with complementary angles. Some students also mishandle the fraction when multiplying across the equation, leading to incorrect coefficients. Writing the equation carefully from the English description is the key step.

Final Answer:
The angle whose measure is two-thirds of its supplementary angle is 72°.