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

Answer 1

A triangle can exist under the following angle conditions:

1. **3 acute angles**: Yes, a triangle can exist with all three angles being acute (less than 90 degrees). The sum of the angles would be 180 degrees.

2. **2 acute angles, 1 right angle**: Yes, a triangle can exist with two acute angles and one right angle (exactly 90 degrees). The sum of the angles would be 180 degrees.

3. **1 acute angle, 1 right angle, 1 obtuse angle**: No, a triangle cannot exist with one acute angle, one right angle, and one obtuse angle. The obtuse angle (greater than 90 degrees) would push the total angle sum beyond 180 degrees.

4. **1 acute angle, 2 obtuse angles**: No, a triangle cannot exist with one acute angle and two obtuse angles. The sum of the angles would also exceed 180 degrees.

5. **2 acute angles, 1 obtuse angle**: No, a triangle cannot exist with two acute angles and one obtuse angle. The obtuse angle already exceeds 90 degrees, making it impossible for the angles to sum to only 180 degrees.

In conclusion, the conditions under which a triangle can exist are:

- 3 acute angles
- 2 acute angles, 1 right angle