Question

discuss mathematical thinking determine whether the statement is always, sometimes, or never true. a triangle has more than one vertex with an acute exterior angle. explain your reasoning. always sometimes never

Explanation:

Step1: Understand the property of exterior angles

An exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Analyze the nature of exterior angles

An acute angle is an angle less than \(90^{\circ}\). A right triangle has two acute interior angles. The exterior angles corresponding to these acute interior angles will be \(90^{\circ}+\) the other non - adjacent interior angle (for the right - angled vertex's exterior angle, it is \(180 - 90=90^{\circ}\)). For an obtuse triangle, two of the interior angles are acute. The exterior angles corresponding to these acute interior angles will be \(180^{\circ}-\) (acute interior angle). Since the acute interior angle is less than \(90^{\circ}\), its exterior angle is greater than \(90^{\circ}\). But for the two acute interior angles of any triangle (except a right - angled triangle where one exterior angle is \(90^{\circ}\)), we can find their exterior angles.
Let the interior angles of a triangle be \(A\), \(B\), and \(C\) with \(A + B + C=180^{\circ}\). The exterior angle \(E_A = B + C\), \(E_B=A + C\), \(E_C=A + B\). If \(A\leqslant B\leqslant C\). If \(C\geqslant90^{\circ}\), then \(A + B=E_C\leqslant90^{\circ}\) (when \(C = 90^{\circ}\), \(E_C=90^{\circ}\); when \(C>90^{\circ}\), \(E_C<90^{\circ}\)). But for the other two exterior angles \(E_A\) and \(E_B\) (since \(A<90^{\circ}\) and \(B < 90^{\circ}\)), \(E_A=B + C>90^{\circ}\) and \(E_B=A + C>90^{\circ}\)

Step3: Conclusion

A triangle always has more than one vertex with an acute exterior angle. Because the sum of interior angles of a triangle is \(180^{\circ}\). If one interior angle \(\theta\geqslant90^{\circ}\), then the sum of the other two interior angles \(180^{\circ}-\theta\leqslant90^{\circ}\). The exterior angles of the two non - \(\theta\) interior angles are \(180^{\circ}-\) (non - \(\theta\) interior angles). Since the non - \(\theta\) interior angles are positive and their sum is \(\leqslant90^{\circ}\), their exterior angles are \(>90^{\circ}\)

Answer:

always