The acute angle of a right-angled triangle is in the ratio 2:3. Find the measure of these angles.

Hint: To solve the question, we have to apply the given information of the ratio of acute angles to obtain relation between the two angles of the right-angled triangle. To solve further, the sum of angles of the triangle is \[{{180}^{0}}\] . Thus, the unknown acute angles of the right-angled triangle can be calculated.

Complete step-by-step answer:
Let ABC be a right-angled triangle with right angle at \[\angle B\]
\[\Rightarrow \angle B={{90}^{0}}\] .… (1)
Thus, \[\angle A,\angle C\]are the acute angles of the right-angled triangle.
Given that the ratio of the acute angles of the right-angled triangle is equal to 2:3
Thus, we get
  \[\dfrac{\angle A}{\angle C}=\dfrac{2}{3}\]
\[\Rightarrow \angle A=\dfrac{2}{3}\angle C\] …. (2)
We know that sum of angles of a triangle is equal to \[{{180}^{0}}\]
Thus, we get \[\angle A+\angle B+\angle C={{180}^{0}}\]
By substituting equation (1) and equation (2) in the above equation, we get
\[\begin{align}
  & \dfrac{2}{3}\angle C+{{90}^{0}}+\angle C={{180}^{0}} \\
 & \dfrac{2}{3}\angle C+\angle C={{180}^{0}}-{{90}^{0}} \\
\end{align}\]
\[\begin{align}
  & \left( \dfrac{2}{3}+1 \right)\angle C={{90}^{0}} \\
 & \left( \dfrac{2+3}{3} \right)\angle C={{90}^{0}} \\
 & \dfrac{5}{3}\angle C={{90}^{0}} \\
 & \Rightarrow \angle C=\dfrac{3}{5}\times {{90}^{0}}=3\times {{18}^{0}}={{54}^{0}} \\
\end{align}\]
Thus, the measurement of acute angle \[\angle C\] is equal to \[{{54}^{0}}\]
By substituting equation (2) in the above equation, we get
\[\angle A=\dfrac{2}{3}\times {{54}^{0}}=2\times {{18}^{0}}={{36}^{0}}\]
Thus, the measurement of acute angle \[\angle A\] is equal to \[{{36}^{0}}\]
Thus, the acute angles of the right-angled triangle are \[{{54}^{0}},{{36}^{0}}\]

Note: The possibility of mistake can be not analysing the given information of the ratio of acute angles given. The possibility of mistake is not applying the sum of angles of the triangle is \[{{180}^{0}}\] . The alternative way of solving is by using a hit-trial method, substitute different angles for the given ratio of angles and apply the sum of angles of the triangle is equal to \[{{180}^{0}}\] . Thus, we can calculate the required answer.