Question

which set of angles can form a triangle? 2 acute and 1 right 1 acute and 2 right 1 acute and 2 obtuse 1 acute, 1 right, and 1 obtuse

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. An acute angle is less than 90°, a right - angle is 90°, and an obtuse angle is greater than 90° and less than 180°.

Step2: Analyze each option

• Option "2 acute and 1 right": Let the acute angles be \(a\) and \(b\) (\(a<90^{\circ}\), \(b < 90^{\circ}\)) and the right - angle be \(c = 90^{\circ}\). Then \(a + b+90^{\circ}<90^{\circ}+90^{\circ}+90^{\circ}=270^{\circ}\) and \(a + b+90^{\circ}=180^{\circ}\) is possible. For example, if \(a = 30^{\circ}\) and \(b = 60^{\circ}\), \(30^{\circ}+60^{\circ}+90^{\circ}=180^{\circ}\).
• Option "1 acute and 2 right": A right - angle is 90°. Two right - angles sum to \(90^{\circ}+90^{\circ}=180^{\circ}\). Adding an acute angle (which is positive) will make the sum of the three angles greater than 180°, so this cannot form a triangle.
• Option "1 acute and 2 obtuse": An obtuse angle is greater than 90°. Two obtuse angles sum to more than \(90^{\circ}+90^{\circ}=180^{\circ}\). Adding an acute angle (which is positive) will make the sum of the three angles greater than 180°, so this cannot form a triangle.
• Option "1 acute, 1 right, and 1 obtuse": A right - angle is 90° and an obtuse angle is greater than 90°. The sum of a right - angle and an obtuse angle is already greater than 180° when adding an acute angle, so this cannot form a triangle.

Answer:

2 acute and 1 right