Question

determining triangle existence using angle measure
under which angle conditions could a triangle exist? choose three correct answers.
2 acute angles, 1 obtuse angle
1 acute angle, 2 obtuse angles
3 acute angles
1 acute angle, 1 right angle, 1 obtuse angle
2 acute angles, 1 right angle

Explanation:

• For "2 acute angles, 1 obtuse angle": The sum of angles in a triangle is \(180^{\circ}\). An acute angle is less than \(90^{\circ}\), and an obtuse angle is greater than \(90^{\circ}\) but less than \(180^{\circ}\). For example, if angles are \(30^{\circ}, 40^{\circ}, 110^{\circ}\), \(30 + 40+110=180^{\circ}\).
• For "3 acute angles": All angles less than \(90^{\circ}\). For example, \(60^{\circ}, 60^{\circ}, 60^{\circ}\), \(60 + 60+60 = 180^{\circ}\).
• For "2 acute angles, 1 right angle": A right - angle is \(90^{\circ}\). Acute angles are less than \(90^{\circ}\). For example, \(30^{\circ}, 60^{\circ}, 90^{\circ}\), \(30 + 60+90=180^{\circ}\).
• For "1 acute angle, 2 obtuse angles": An obtuse angle \(>90^{\circ}\). The sum of two obtuse angles \(>180^{\circ}\). So, it's impossible.
• For "1 acute angle, 1 right angle, 1 obtuse angle": The sum of a right - angle (\(90^{\circ}\)) and an obtuse angle (\(>90^{\circ}\)) already \(>180^{\circ}\). So, it's impossible.

Answer:

2 acute angles, 1 obtuse angle; 3 acute angles; 2 acute angles, 1 right angle.