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 = 90°, m∠2 = 90°. (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 = 0°.

Explanation:

Step1: Recall angle - type definitions

An obtuse angle is an angle with measure greater than 90° and less than 180°. Supplementary angles are two angles whose sum is 180°. Complementary angles are two angles whose sum is 90°.

Step2: Provide counter - example for part (a)

If \(m\angle1 = 90^{\circ}\) and \(m\angle2=90^{\circ}\), \(\angle1\) and \(\angle2\) are supplementary (\(90 + 90=180\)), but neither is obtuse since they are right angles.

Step3: Provide counter - example for part (b)

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

Answer:

(a) Counterexample: \(m\angle1 = 90^{\circ}\), \(m\angle2 = 90^{\circ}\)
(b) Counterexample: \(m\angle1 = 45^{\circ}\), \(m\angle2 = 45^{\circ}\)