Question

which is a counterexample that shows the following conjecture is false: \if ∠1 and ∠2 are supplementary, then one of the angles is obtuse\? o m∠1 = 100° and m∠2 = 80°. o m∠1 = 45° and m∠2 = 45°. o m∠1 = 90° and m∠2 = 90°. o m∠1 = 53° and m∠2 = 127°.

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles add up to 180°. An obtuse - angle is an angle with a measure greater than 90° and less than 180°.

Step2: Analyze each option

For the pair where \(m\angle1 = 45^{\circ}\) and \(m\angle2=45^{\circ}\), \(m\angle1 + m\angle2=45^{\circ}+45^{\circ}=90^{\circ}
eq180^{\circ}\), so this pair is not supplementary.
For the pair where \(m\angle1 = 90^{\circ}\) and \(m\angle2 = 90^{\circ}\), \(m\angle1 + m\angle2=90^{\circ}+90^{\circ}=180^{\circ}\), but neither angle is obtuse.
For the pair where \(m\angle1 = 53^{\circ}\) and \(m\angle2 = 127^{\circ}\), \(m\angle1 + m\angle2=53^{\circ}+127^{\circ}=180^{\circ}\) and \(\angle2\) is obtuse.
For the pair where \(m\angle1 = 100^{\circ}\) and \(m\angle2 = 80^{\circ}\), \(m\angle1 + m\angle2=100^{\circ}+80^{\circ}=180^{\circ}\) and \(\angle1\) is obtuse.

The counter - example should be a pair of supplementary angles where neither is obtuse. The pair \(m\angle1 = 90^{\circ}\) and \(m\angle2 = 90^{\circ}\) is a counter - example as they are supplementary (sum to 180°) but neither is obtuse.

Answer:

\(m\angle1 = 90^{\circ}\) and \(m\angle2 = 90^{\circ}\)