Question

critical thinking tell whether the statement is always, sometimes, or never true. always sometimes never if two complementary angles are congruent, then the measure of each angle is 45°. explain your reasoning. the sum of 90° divided by 2 is 45°. there are an infinite number of pairs of angle measures that sum to 90°. there are an infinite number of pairs of angle measures that sum to 180°. the sum of 180° divided by 2 is 90°, not 45°.

Explanation:

Step1: Recall complementary angles

Complementary angles sum to \(90^\circ\). Let the two congruent complementary angles be \(x\) and \(x\) (since they are congruent, their measures are equal).

Step2: Set up the equation

The sum of complementary angles is \(90^\circ\), so \(x + x=90^\circ\).

Step3: Solve the equation

Simplify the left - hand side: \(2x = 90^\circ\). Then divide both sides by 2: \(x=\frac{90^\circ}{2}=45^\circ\). So if two complementary angles are congruent, each must be \(45^\circ\), which means the statement is always true.
For the reasoning part, the first option "The sum of \(90^\circ\) divided by 2 is \(45^\circ\)" is correct because we derived that each angle is \(\frac{90^\circ}{2} = 45^\circ\) from the definition of complementary angles and the fact that the two angles are congruent. The second option is about non - congruent complementary angles, the third is about supplementary angles (sum to \(180^\circ\)) which is not relevant here, and the fourth is also about supplementary angles and incorrect for this problem.

Answer:

The statement "If two complementary angles are congruent, then the measure of each angle is \(45^\circ\)" is always true. The correct reasoning is "The sum of \(90^\circ\) divided by 2 is \(45^\circ\)".