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:

Step1: Recall triangle angle sum

Sum of angles = $180^\circ$. Acute < $90^\circ$, right = $90^\circ$, obtuse > $90^\circ$.

Step2: Check 3 acute angles

E.g., 60°+60°+60°=180°, valid.

Step3: Check 2 acute +1 right

90°+45°+45°=180°, valid.

Step4: Check 1 acute+1 right+1 obtuse

Right+obtuse >90°+90°=180°, invalid.

Step5: Check 1 acute+2 obtuse

2 obtuse >90°×2=180°, invalid.

Step6: Check 2 acute+1 obtuse

120°+30°+30°=180°, valid.

Answer:

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