Question

If ABC is an obtuse triangle with $\mathrm{\angle }B$ as the obtuse angle, then what is the measure of $\mathrm{\angle }B$? and measure of $\mathrm{\angle }A$ and measure of $\mathrm{\angle }C$ is $<{90}^0$.

Answer 1

Hint: According to the given question we have to find the measure of obtuse angle $\mathrm{\angle }B$ for the given obtuse $\mathrm{\Delta }$ABC but first of all we have to understand the obtuse angle.
Obtuse angle: An obtuse angle is the angle which is more than ${90}^0(>{90}^0)$ but less than ${180}^0(<{180}^0)$ we can also say that an obtuse angle is the angle which is between right angle $({90}^0)$ and the straight line that makes (${180}^0$) angle. We can also understand it with the help of the diagram given below:

As we can see in the diagram given above, $\mathrm{\angle }ABC$ is an obtuse angle which is between the right angle $\mathrm{\angle }SBC={90}^0$ and the straight line $\mathrm{\angle }TBC={180}^0$

Complete step-by-step solution:
Given,
$\mathrm{\Delta }$ABC is an obtuse triangle and,
Measures of $\mathrm{\angle }A$ and measure of $\mathrm{\angle }C$ is $<{90}^0$
Now, we have to measure the obtuse angle $\mathrm{\angle }B$ for the given obtuse $\mathrm{\Delta }$ABC we have to understand about the obtuse triangle.
Obtuse triangle: An obtuse triangle is the triangle in which it’s one interior angle measure is more than the ${90}^0$ and if one angle measures more than ${90}^0$, then the sum of it’s remaining two interior angles is less than ${90}^0$. We can also understand it with the help of the diagram given below:

Hence, for the given triangle ABC, $\mathrm{\angle }ABC>{90}^0$
Step 1: To measure the obtuse angle $\mathrm{\angle }B$ (An obtuse angle is the angle which is more than ${90}^0(>{90}^0)$ but less than ${180}^0(<{180}^0)$) for the given obtuse triangle ABC (An obtuse triangle is the triangle in which it’s one interior angle measure is more than the ${90}^0$and if one angle measures more than ${90}^0$, then the sum of its remaining two interior angles is less than ${90}^0$)
We have to let that the given obtuse angle is $\mathrm{\angle }B={120}^0$(which is an obtuse angle).
Step 2: As we know, the sum of all the interior angles of a triangle is ${180}^0$. So, for the given triangle ABC,

$\mathrm{\angle }ABC+\mathrm{\angle }BAC+\mathrm{\angle }ACB={180}^0$……………………..(1)
Step 3: As we let that an obtuse angle $\mathrm{\angle }B={120}^0$ so, on substituting the value of obtuse angle $\mathrm{\angle }B$ in equation (1).
$$\begin{gathered} {120}^0+\mathrm{\angle }BAC+\mathrm{\angle }ACB={180}^0 \\ \mathrm{\angle }BAC+\mathrm{\angle }ACB={180}^0-{120}^0 \\ \mathrm{\angle }BAC+\mathrm{\angle }ACB={60}^0 \end{gathered}$$
So, measure of $\mathrm{\angle }BAC$ and measure of $\mathrm{\angle }ACB$ is $<{90}^0$ and measure of $\mathrm{\angle }ABC$ is more than ${90}^0$
Hence, we can say that for a obtuse triangle ABC, obtuse angle $\mathrm{\angle }B$ is more than ${90}^0$ and the measure of $\mathrm{\angle }A$ and measure of $\mathrm{\angle }C$ is $<{90}^0$.

Note:
1. The side opposite to the obtuse angle is the longest side for that given triangle.
2. An equilateral triangle can’t be an obtuse triangle because it’s all sides are equal in size and it’s all the interior angles are ${60}^0$ in measure.
3. A triangle can’t be right-angled and obtuse angled at the same time. Since a right angle has one right angle and the other two interior angles are acute.