Question

under which angle conditions could a triangle exist? check all that apply.
3 acute angles
2 acute angles, 1 right angle
1 acute angle, 1 right angle, 1 obtuse angle
1 acute angle, 2 obtuse angles
2 acute angles, 1 obtuse angle

Explanation:

• 3 acute angles: A triangle with three acute angles (each less than \(90^{\circ}\)) is called an acute - angled triangle. Since the sum of the interior angles of a triangle is \(180^{\circ}\), and if each angle \(\alpha<90^{\circ}\), \(\beta < 90^{\circ}\), \(\gamma<90^{\circ}\), then \(\alpha+\beta+\gamma<270^{\circ}\). But when \(\alpha+\beta+\gamma = 180^{\circ}\) (e.g., \(\alpha = 60^{\circ}\), \(\beta=60^{\circ}\), \(\gamma = 60^{\circ}\)), such a triangle exists.
• 2 acute angles, 1 right angle: A right - angled triangle has one angle equal to \(90^{\circ}\). Let the other two angles be \(\alpha\) and \(\beta\). By the angle - sum property of a triangle \(\alpha+\beta+90^{\circ}=180^{\circ}\), so \(\alpha+\beta = 90^{\circ}\). Since \(\alpha>0\) and \(\beta>0\), \(\alpha<90^{\circ}\) and \(\beta<90^{\circ}\) (they are acute). For example, \(\alpha = 30^{\circ}\), \(\beta=60^{\circ}\), and the right angle is \(90^{\circ}\).
• 1 acute angle, 1 right angle, 1 obtuse angle: The sum of a right angle (\(90^{\circ}\)) and an obtuse angle (\(\theta>90^{\circ}\)) is \(90^{\circ}+\theta>180^{\circ}\). Let the third angle be \(\alpha\) (acute, \(\alpha<90^{\circ}\)). Then \(\alpha + 90^{\circ}+\theta>180^{\circ}\), which violates the angle - sum property of a triangle (\(\alpha+\beta+\gamma=180^{\circ}\)).
• 1 acute angle, 2 obtuse angles: Let the two obtuse angles be \(\theta_1>90^{\circ}\) and \(\theta_2>90^{\circ}\). Then \(\theta_1+\theta_2>180^{\circ}\). Let the third angle be \(\alpha\) (acute, \(\alpha < 90^{\circ}\)). So \(\alpha+\theta_1+\theta_2>180^{\circ}\), which violates the angle - sum property of a triangle.
• 2 acute angles, 1 obtuse angle: Let the obtuse angle be \(\theta>90^{\circ}\). Let the other two angles be \(\alpha\) and \(\beta\). By the angle - sum property \(\alpha+\beta+\theta=180^{\circ}\), so \(\alpha+\beta=180^{\circ}-\theta\). Since \(\theta>90^{\circ}\), \(180^{\circ}-\theta < 90^{\circ}\). Also, \(\alpha>0\) and \(\beta>0\), so \(\alpha<90^{\circ}\) and \(\beta<90^{\circ}\) (they are acute). For example, \(\alpha = 30^{\circ}\), \(\beta = 40^{\circ}\), \(\theta=110^{\circ}\)

Answer:

• 3 acute angles
• 2 acute angles, 1 right angle
• 2 acute angles, 1 obtuse angle