What is the maximum number of obtuse angles a triangle can have? A. 2 B. 0 C. 1 D. 3

To determine the maximum number of obtuse angles a triangle can have, we need to consider the properties of triangles and the definition of obtuse angles. 1. **Definition of an obtuse angle**: An obtuse angle is an angle that measures more than 90 degrees but less than 180 degrees. 2. **Sum of angles in a triangle**: The sum of the interior angles of any triangle is always 180 degrees. Now, let's analyze the options: - If a triangle had **3 obtuse angles**, the minimum sum of the angles would be \(3 \times 91 = 273\) degrees, which exceeds 180 degrees. Therefore, a triangle cannot have 3 obtuse angles. - If a triangle had **2 obtuse angles**, the minimum sum of those two angles would be \(2 \times 91 = 182\) degrees. This also exceeds 180 degrees, so a triangle cannot have 2 obtuse angles. - If a triangle had **1 obtuse angle**, the maximum it could be is 179 degrees. The remaining angle would then be \(180 - 179 = 1\) degree, which is valid. Thus, a triangle can have 1 obtuse angle. - A triangle cannot have **0 obtuse angles** if it has at least one angle, but it can have all angles acute (less than 90 degrees). Based on this analysis, the maximum number of obtuse angles a triangle can have is: **C. 1**