Question

show that the conjecture is false by providing a counterexample.
if the measures of \\( \angle r, \angle s \\), and \\( \angle t \\) sum to \\( 180 ^ { \circ } \\), then one of the angles must be obtuse.
counterexample: \\( m \angle r = \square ^ { \circ }, m \angle s = \square ^ { \circ }, m \angle t = \square ^ { \circ } \\)

Explanation:

Step1: Recall the definition of an obtuse angle

An obtuse angle is an angle whose measure is greater than \(90^{\circ}\) and less than \(180^{\circ}\). We need to find three angles \(\angle R\), \(\angle S\), \(\angle T\) such that \(m\angle R + m\angle S+m\angle T = 180^{\circ}\) and none of the angles is obtuse.

Step2: Choose non - obtuse angles

Let's consider three acute angles. An acute angle is an angle whose measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\). For example, if \(m\angle R=60^{\circ}\), \(m\angle S = 60^{\circ}\), and \(m\angle T=60^{\circ}\). Then \(m\angle R + m\angle S+m\angle T=60 + 60+60=180^{\circ}\), and each angle (\(60^{\circ}\)) is acute (not obtuse).

Answer:

\(m\angle R = 60^{\circ}\), \(m\angle S = 60^{\circ}\), \(m\angle T = 60^{\circ}\)