Question

How many acute angles can a right triangle have?

Answer 1

Hint: Here we have to determine the number of possible acute angles in a right-angled triangle. We know that a triangle is defined as a regular polygon, which has three sides and the sum of any two sides of the triangle is always greater than the third side. We will use the definition of the right angle triangle here and also the important properties of the triangle to get the required answer.

Complete step by step solution:
Here we have to determine the number of possible acute angles in a right-angle triangle.
A right-angled triangle is one of the types of triangles that has 3 sides’ i.e. the base, hypotenuse, and the perpendicular in which the angle between the base and height is equal to $90{}^{\circ }$.
We will draw the triangle $ABC$ which is right-angled at $B$.

We also know that the sum of all the angles of the triangle is equal to $180{}^{\circ }$.
We will sue this property here in this triangle.
$\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }$
We know that the value of $\mathrm{\angle }B={90}^{\circ }$. So we will substitute the value here.
$\Rightarrow \mathrm{\angle }A+{90}^{\circ }+\mathrm{\angle }C={180}^{\circ }$
Now, we will subtract ${90}^{\circ }$ from both sides.
$$\begin{gathered} \Rightarrow \mathrm{\angle }A+{90}^{\circ }+\mathrm{\angle }C-{90}^{\circ }={180}^{\circ }-{90}^{\circ } \\ \Rightarrow \mathrm{\angle }A+\mathrm{\angle }C={90}^{\circ } \end{gathered}$$
Thus, both of the remaining two angles must have a measure less than ${90}^{\circ }$ and therefore must be acute.

Hence, the right triangle can have only two acute angles.

Note:
Here we have obtained the number of possible acute angles in a right-angled triangle. We know that acute angles are the angles that are less than ${90}^{\circ }$, whereas the obtuse angle is an angle that is greater than ${90}^{\circ }$. We also know that a straight line makes ${180}^{\circ }$ whereas the reflex angle is an angle that is greater than ${1800}^{\circ }$ but less than ${360}^{\circ }$. In an acute angle triangle, all the angles of the triangle are acute. In an obtuse angle triangle, one of the three angles is obtuse the other two are acute angles.