under which angle conditions could a triangle exist? check all that apply.\n3 acute angles\n2 acute angles…

under which angle conditions could a triangle exist? check all that apply.\n3 acute angles\n2 acute angles, 1 right angle\n1 acute angle, 1 right angle, 1 obtuse angle\n1 acute angle, 2 obtuse angles\n2 acute angles, 1 obtuse angle

under which angle conditions could a triangle exist? check all that apply.\n3 acute angles\n2 acute angles, 1 right angle\n1 acute angle, 1 right angle, 1 obtuse angle\n1 acute angle, 2 obtuse angles\n2 acute angles, 1 obtuse angle

Answer

Explanation:

Step1: Recall the sum of angles in a triangle

The sum of the interior angles of a triangle 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}) but less than (180^{\circ}).

Step2: Analyze each option

  • Option 1: 3 acute angles Let the angles be (a), (b), (c) where (a<90^{\circ}), (b < 90^{\circ}), (c<90^{\circ}). Then (a + b + c<270^{\circ}). For example, (a = 60^{\circ}), (b=60^{\circ}), (c = 60^{\circ}), (a + b + c=180^{\circ}). So a triangle with 3 acute angles (an acute - angled triangle) can exist.
  • Option 2: 2 acute angles, 1 right angle Let the acute angles be (x) and (y) ((x<90^{\circ}), (y < 90^{\circ})) and the right - angle (z = 90^{\circ}). Then (x + y+z=x + y + 90^{\circ}). Since (x+y=180^{\circ}-z), and (x,y>0^{\circ}), for example (x = 30^{\circ}), (y = 60^{\circ}), (z = 90^{\circ}), (x + y+z=180^{\circ}). So a right - angled triangle (with 2 acute and 1 right angle) can exist.
  • Option 3: 1 acute angle, 1 right angle, 1 obtuse angle Let the acute angle (a<90^{\circ}), the right angle (b = 90^{\circ}), and the obtuse angle (c>90^{\circ}). Then (a + b + c>90^{\circ}+90^{\circ}+0^{\circ}=180^{\circ}). So a triangle with 1 acute, 1 right, and 1 obtuse angle cannot exist.
  • Option 4: 1 acute angle, 2 obtuse angles Let the acute angle (a<90^{\circ}) and the obtuse angles (b>90^{\circ}), (c>90^{\circ}). Then (a + b + c>90^{\circ}+90^{\circ}+0^{\circ}=180^{\circ}). So a triangle with 1 acute and 2 obtuse angles cannot exist.
  • Option 5: 2 acute angles, 1 obtuse angle Let the acute angles be (x) and (y) ((x<90^{\circ}), (y < 90^{\circ})) and the obtuse angle (z>90^{\circ}). Then (x + y+z). Since (x + y=180^{\circ}-z), and (x,y>0^{\circ}), for example (x = 30^{\circ}), (y = 40^{\circ}), (z = 110^{\circ}), (x + y+z=180^{\circ}). So a triangle with 2 acute and 1 obtuse angle (an obtuse - angled triangle) can exist.

Answer:

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