An angle is smaller than its supplementary angle by 30°. What is the measure of this angle in degrees?

Introduction / Context: This problem checks understanding of supplementary angles and how to translate a verbal relationship into a simple algebraic equation. Supplementary angle problems are very common in basic geometry and aptitude tests.

Given Data / Assumptions: - Two angles are supplementary if their sum is 180°. - One of these angles is smaller than its supplementary angle by 30°. - We are asked for the measure of this smaller angle.

Concept / Approach: Let the smaller angle be x degrees. Its supplementary angle must then be 180° − x degrees. The statement says that the smaller angle is 30° less than the supplementary angle. That means: x = (180° − x) − 30°. We can solve this linear equation for x to find the required angle.

Step-by-Step Solution: Step 1: Let the smaller angle be x°. Step 2: Its supplementary angle is 180° − x°. Step 3: According to the question, x° is 30° less than its supplementary angle, so write x = (180° − x) − 30°. Step 4: Simplify the right side: (180° − x) − 30° = 150° − x. Step 5: The equation is x = 150° − x. Step 6: Add x to both sides: x + x = 150°, so 2x = 150°. Step 7: Divide both sides by 2: x = 75°.

Verification / Alternative Check: If the smaller angle is 75°, then its supplementary angle is 180° − 75° = 105°. The smaller angle is indeed 30° less than the supplementary angle since 105° − 75° = 30°. This confirms that 75° is correct.

Why Other Options Are Wrong: Option b, 105°, is actually the larger supplementary angle, not the smaller one. Option c, 60°, would have a supplementary angle of 120°, giving a difference of 60°, not 30°. Option d, 30°, would result in a supplementary angle of 150°, with a difference of 120°, which does not match the given condition.

Common Pitfalls: A common error is setting up the equation with the roles reversed or accidentally using 90° instead of 180° by confusing supplementary and complementary angles. Another pitfall is misinterpreting the phrase smaller than its supplementary angle by 30° and writing 180° − x = x − 30° instead of the correct relation. Carefully reading and translating the language into algebra ensures the correct equation.

Final Answer: The measure of the smaller angle is 75°.