Question

find one counterexample to show that the conjecture is false. ∠1 and ∠2 are supplementary, so one of the angles is acute. choose the best counterexample for the conjecture. a. ( mangle 1 = 109^circ ) and ( mangle 2 = 71^circ ) b. ( mangle 1 = 90^circ ) and ( mangle 2 = 90^circ ) (partially visible text)

Explanation:

Step1: Recall supplementary angles and acute angles

Supplementary angles sum to \(180^\circ\). An acute angle is less than \(90^\circ\), an obtuse angle is greater than \(90^\circ\) and less than \(180^\circ\), and a right angle is \(90^\circ\). We need a case where both angles are not acute (i.e., at least one is non - acute, but to disprove "one of the angles is acute", we can have both non - acute, like right angles or obtuse angles, but right angles sum to \(180^\circ\) when both are \(90^\circ\)).

Step2: Analyze option B (assuming option B is \(m\angle1 = 90^\circ\) and \(m\angle2=90^\circ\))

First, check if they are supplementary: \(m\angle1 + m\angle2=90^\circ + 90^\circ = 180^\circ\), so they are supplementary. Now, check if any is acute. An acute angle is less than \(90^\circ\), but both angles are \(90^\circ\) (right angles, not acute). So this is a counterexample because the two angles are supplementary, but neither is acute, which shows the conjecture " \(\angle1\) and \(\angle2\) are supplementary, so one of the angles is acute" is false.

Step3: Analyze option A

For option A, \(m\angle1 = 109^\circ\) (obtuse) and \(m\angle2 = 71^\circ\) (acute). Here, one angle is acute, so this does not counter the conjecture.

Answer:

B. \(m\angle1 = 90^\circ\) and \(m\angle2 = 90^\circ\)