Question

Thinking Critically about Supplementary Angles
Is the following statement always, sometimes, or never true?

“The supplement of an acute angle is an obtuse angle.”

Explain your answer.

Answer 1

Always true.

Proof: Let A be an acute angle, so 0° < A < 90°. Its supplement S satisfies S = 180° − A. Then
180° − 90° < 180° − A < 180° − 0°,
so 90° < S < 180°. Hence S is an obtuse angle. (Same argument in radians: 0 < θ < π/2 ⇒ π/2 < π − θ < π.)