Question

show that each conjecture is false by providing a counterexample. (a) if ∠1 and ∠2 are supplementary angles, then one of them must be obtuse. counterexample: m∠1 = 0°, m∠2 = °. (b) if ∠1 and ∠2 are complementary angles, then one of them must have a measure less than 45°. counterexample: m∠1 = 0°, m∠2 = °.

Explanation:

Step1: Recall angle - sum definitions

Supplementary angles add up to 180°. Complementary angles add up to 90°.

Step2: Find counter - example for supplementary angles

Let \(m\angle1 = 90^{\circ}\) and \(m\angle2=90^{\circ}\). Since \(90 + 90=180\), they are supplementary, but neither is obtuse (an obtuse angle has a measure between 90° and 180°).

Step3: Find counter - example for complementary angles

Let \(m\angle1 = 45^{\circ}\) and \(m\angle2 = 45^{\circ}\). Since \(45+45 = 90\), they are complementary, and neither has a measure less than 45°.

Answer:

(a) \(m\angle1 = 90\), \(m\angle2 = 90\)
(b) \(m\angle1 = 45\), \(m\angle2 = 45\)