Question

critical thinking tell whether the statement is always, sometimes, or never true.
always sometimes never
if an angle is acute, then its complement is greater than its supplement.
explain your reasoning.
its complement must be obtuse, and its supplement must be acute.
its complement must be greater than its supplement when the angle is less than 45°.
its complement must be greater than its supplement when the angle is greater than 90°.
its complement must be acute, and its supplement must be obtuse.

Explanation:

Step1: Recall angle definitions

An acute angle \( \theta \) satisfies \( 0^\circ < \theta < 90^\circ \). The complement of \( \theta \) is \( 90^\circ - \theta \), and the supplement is \( 180^\circ - \theta \).

Step2: Compare complement and supplement

Calculate the difference between supplement and complement: \( (180^\circ - \theta) - (90^\circ - \theta) = 90^\circ \). So the supplement is always \( 90^\circ \) greater than the complement. Thus, the complement is never greater than the supplement for an acute angle.

Step3: Analyze reasoning options

• First option: Complement of acute \( \theta \) is \( 90^\circ - \theta \) (acute, since \( \theta < 90^\circ \)), supplement is \( 180^\circ - \theta \) (obtuse, since \( \theta < 90^\circ \)). So this is wrong.
• Second option: As shown, supplement is always \( 90^\circ \) more, so complement can't be greater, even for \( \theta < 45^\circ \). Wrong.
• Third option: Acute angles are less than \( 90^\circ \), so this is invalid. Wrong.
• Fourth option: Correctly states complement (acute) and supplement (obtuse), and since supplement is always larger, the original statement is never true.

Answer:

Never; The reasoning is: Its complement must be acute, and its supplement must be obtuse. (And since supplement \( = 90^\circ + \) complement, complement is never greater than supplement for an acute angle.)