In a right angled triangle, the other two angles are always

Question

Accepted Answer

The correct option is 1:
In a right-angled triangle, one angle is exactly $90^\circ$. A fundamental property of all triangles is that the sum of their interior angles is always $180^\circ$. Let the three angles of a right-angled triangle be $\angle A$, $\angle B$, and $\angle C$. If we assume $\angle A$ is the right angle, then $\angle A = 90^\circ$.

According to the angle sum property of a triangle:
$\angle A + \angle B + \angle C = 180^\circ$
Substituting the value of $\angle A$:
$90^\circ + \angle B + \angle C = 180^\circ$
Subtract $90^\circ$ from both sides to find the sum of the other two angles:
$\angle B + \angle C = 180^\circ - 90^\circ$
$\angle B + \angle C = 90^\circ$
Since the sum of $\angle B$ and $\angle C$ is $90^\circ$, neither of these angles can be $90^\circ$ or greater, because that would make the sum exceed $90^\circ$ (or make one angle zero, which is not possible in a triangle). For example, if $\angle B = 90^\circ$, then $\angle C$ would have to be $0^\circ$, which is not possible. Similarly, if $\angle B$ were obtuse (greater than $90^\circ$), then $\angle C$ would have to be a negative angle, which is impossible. Therefore, both $\angle B$ and $\angle C$ must be less than $90^\circ$. An angle that measures less than $90^\circ$ is called an acute angle. Thus, the other two angles in a right-angled triangle are always acute.

The incorrect option is 2:
If one of the other two angles were a right angle ($90^\circ$), then the sum of the two angles alone would be $90^\circ + 90^\circ = 180^\circ$. This would leave $0^\circ$ for the third angle, which is impossible in a triangle. A triangle cannot have two right angles.

The incorrect option is 3:
If one of the other two angles were obtuse (greater than $90^\circ$), then the sum of that obtuse angle and the right angle ($90^\circ$) would already exceed $180^\circ$. For example, if an angle is $100^\circ$, then $100^\circ + 90^\circ = 190^\circ$. This is greater than the total sum of angles allowed in a triangle ($180^\circ$), making it impossible for the third angle to exist. A triangle cannot have an obtuse angle and a right angle simultaneously.

The incorrect option is 4:
Since we have determined that the other two angles must always be acute, "None of the above" is an incorrect choice.