Question

In a triangle ABC, if the angle B is obtuse, what types of angles can angle A and C be?

Answer 1

Hint: In the above question, we have to find the measure of remaining two angles. We will take several conditions using the properties of right-angle and obtuse angle to get to the final answer. We also know that the sum of all the angles of a triangle is ${180^ \circ }$.

Complete step by step answer:
In the above question, we know that sum of all the angles of a triangle is ${180^ \circ }$.An obtuse angle is more than ${90^ \circ }$, so the sum of the remaining $2$ angles have to be less than ${90^ \circ }$.Now there are some conditions, note that it is impossible to have:
$\left( 1 \right)$ $2$ right angles in a triangle, because ${90^ \circ } + {90^ \circ } = {180^ \circ }$ and the third angle still needs to be added.
$\left( 2 \right)$ $1$ obtuse and $1$ right angle in a triangle, their sum is more than ${180^ \circ }$.
$\left( 3 \right)$ $2$ obtuse angles in a triangle, their sum is more than ${180^ \circ }$.
It is possible to have an obtuse-angled isosceles triangle, but the vertex angle must be obtuse and the equal base angles will be acute.

Therefore, from considering the above possibilities it is clear that angle A and C are acute.

Note: The obtuse angle of a triangle is a triangle, where one of its angles of a triangle is greater than ${90^ \circ }$ degrees. An obtuse triangle may be an isosceles or scalene triangle. An equilateral triangle cannot be obtuse. The angle opposite to the obtuse angle is the longest side of the triangle. Similarly, a triangle cannot have a right angle and obtuse angle at the same time. If one of the angles of a triangle is obtuse, the other two angles of a triangle must be acute.