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:

• For a triangle, the sum of interior angles is \(180^{\circ}\).
• An acute angle is less than \(90^{\circ}\), a right angle is equal to \(90^{\circ}\), and an obtuse angle is greater than \(90^{\circ}\).
• Option 1: If there are 3 acute angles (e.g., \(60^{\circ},60^{\circ},60^{\circ}\)), their sum can be \(180^{\circ}\).
• Option 2: If there are 2 acute angles and 1 right angle (e.g., \(45^{\circ},45^{\circ},90^{\circ}\)), their sum can be \(180^{\circ}\).
• Option 3: If there is 1 acute angle (\(<90^{\circ}\)), 1 right angle (\( = 90^{\circ}\)), and 1 obtuse angle (\(>90^{\circ}\)), the sum will be \(>180^{\circ}\).
• Option 4: If there is 1 acute angle (\(<90^{\circ}\)) and 2 obtuse angles (\(>90^{\circ}\) each), the sum will be \(>180^{\circ}\).
• Option 5: If there are 2 acute angles and 1 obtuse angle (e.g., \(30^{\circ},30^{\circ},120^{\circ}\)), their sum can be \(180^{\circ}\).

Answer:

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