 QUESTION

# The Range of an obtuse angle is:a. $\left[ {{0}^{\circ }},{{90}^{\circ }} \right)$b. $\left( {{0}^{\circ }},{{90}^{\circ }} \right]$c. $\left[ {{90}^{\circ }},{{180}^{\circ }} \right)$d. $\left( {{90}^{\circ }},{{180}^{\circ }} \right)$

Hint: Find the angle which forms the obtuse angle and decide the bracket which denotes the range specified by obtuse angle, which means that the range includes the listed elements.

We know that acute angle is less than ${{90}^{\circ }}$. And the obtuse angle is greater than ${{90}^{\circ }}$ but less than ${{180}^{\circ }}$.

Obtuse angle does not include the angle ${{90}^{\circ }}$ and ${{180}^{\circ }}$. Thus the angle lies in between ${{90}^{\circ }}$ and ${{180}^{\circ }}$.
Now we need to know how to bracket the values ${{90}^{\circ }}$ and ${{180}^{\circ }}$.
A bracket means that the end of the range is inclusive- it includes the element listed. A parenthesis means that the end is exclusive and doesn’t contain the listed element. So for [ first 1, Last 1), the range starts with first 1 (and includes it), but ends just before last 1.

Let us take some examples,
(0, 5) = 1, 2, 3, 4 : 0 and 5 are not included.
(0, 5] = 1, 2, 3, 4, 5 : 0 is not included.
[0, 5) = 0, 1, 2, 3, 4 : 5 is not included.
[0, 5] = 0, 1, 2, 3, 4, 5 : All the values in range are included.

We said that an obtuse angle is greater than ${{90}^{\circ }}$ but it is less than ${{180}^{\circ }}$. But it does not include the angle ${{90}^{\circ }}$ and ${{180}^{\circ }}$. Thus we can represent it as $\left( {{90}^{\circ }},{{180}^{\circ }} \right)$.
$\left( {{90}^{\circ }},{{180}^{\circ }} \right)$ all the values will be there except for ${{90}^{\circ }}$ and ${{180}^{\circ }}$.

Thus the range of obtuse angle is $\left( {{90}^{\circ }},{{180}^{\circ }} \right)$.
$\therefore$ Option (d) is the correct answer.

Note: Similarly, we can get the range of an acute angle as $\left( {{90}^{\circ }},{{180}^{\circ }} \right)$, where the acute angle is greater than ${{0}^{\circ }}$ but less than ${{90}^{\circ }}$. Thus in the representation of range $\left( {{90}^{\circ }},{{180}^{\circ }} \right)$ all the angles are included except ${{0}^{\circ }}$ and ${{90}^{\circ }}$.