Question

How many obtuse angles does a triangle have?

Answer 1

Hint: The obtuse angle is the angle which is greater than \[{90^ \circ }\] but it is less than ${180^ \circ }$. The total angle of triangles must add up to ${180^ \circ }$. When the interior angles of the triangle are added they must add up to ${180^ \circ }$.

Complete Step by Step Solution:
According to the question, we have to find the maximum number of obtuse angles that the triangle must-have.
The triangle is the closed two – a dimensional plane figure which has three sides and three angles. On the basis of sides of triangles and angles of triangles, we can form different types of triangles and obtuse angles are one among them.
The obtuse angle can be defined as the angle which is greater than \[{90^ \circ }\] but is less than ${180^ \circ }$. When one of the interior angles of the triangle is less than ${180^ \circ }$ but greater than \[{90^ \circ }\] then, that triangle is termed as an obtuse-angled triangle.
When the triangle is obtuse-angled then there is only one obtuse angle and the other two angles are acute. Acute angles are those which measure less than \[{90^ \circ }\]. When one of the interior angles of the triangle is obtuse then there must be two acute angles because when an angle in the triangle is greater than \[{90^ \circ }\], suppose the triangle has an obtuse angle which is ${91^ \circ }$ as the angles of the triangle must add up to ${180^ \circ }$. So, if one of the angles of the triangle is ${91^ \circ }$ then, ${180^ \circ } - {91^ \circ } = {89^ \circ }$, then, other two angles must add up to ${89^ \circ }$ as the obtuse angle must be greater than \[{90^ \circ }\].

Hence, there can be a maximum of one obtuse angle in the triangle.

Note: The other way of finding the obtuse-angled triangle is when the sum of the squares of smaller sides is less than the square of the largest side. Let a, b and c be the sides of the triangle and c be the largest side, then the triangle is obtuse if –
${a^2} + {b^2} < {c^2}$.