Question

The acute angles of a right angled triangle are in the ratio \[2:3\] . Find the angles of the triangle?

Answer 1

Hint: The triangle is a right angle triangle meaning one of the angle of the triangle is \[{{90}^{\circ }}\] and other two angles are less than \[{{90}^{\circ }}\] as to state the acute angles of the triangles. Now the triangle’s acute angle is divided in ratio of \[2:3\] and the formula to find the angles are:
Sum of all the angles of a triangle is equal to \[{{180}^{\circ }}\] .

Complete step-by-step answer:
Now let us check the question, as we know that the question says that there are two angles (acute) and the ratio of the angle is given as well so to find the two angles let us substitute a random variable to the ratios to convert them into angles. Hence, let us assume the common unknown variable amongst the two is \[x\] and the angles are given as \[2x\] and \[3x\] .
Hence, using the angles in the formula as given in the hint, we take the sum of all the angles of the triangle as:
 \[\Rightarrow 2x+3x+{{90}^{\circ }}={{180}^{\circ }}\]
The triangle is a right angle triangle so the third angle is \[{{90}^{\circ }}\] .
 \[\Rightarrow 5x+{{90}^{\circ }}={{180}^{\circ }}\]
 \[\Rightarrow 5x={{180}^{\circ }}-{{90}^{\circ }}\]
 \[\Rightarrow x={{18}^{\circ }}\]
Now as we have gotten the common unknown variable of the two acute triangle we can find the two angles of the triangle as \[2x\] and \[3x\] where putting the value of \[x={{18}^{\circ }}\] , we can get the value of the two angles as:
 \[2x=2\times 18\] and \[3x=3\times 18\]
 \[36\] and \[54\]
Therefore, the two acute angles of the triangles are given as \[36\] and \[54\] with the third one as \[{{90}^{\circ }}\].
So, the correct answer is “ \[36\] and \[54\] with the third one as \[{{90}^{\circ }}\].”.

Note: Acute angles are angle which are less than \[{{90}^{\circ }}\] and the third angle is \[{{90}^{\circ }}\] meaning that the sum of the two angles that are in ratio of \[2:3\] are equal to \[{{90}^{\circ }}\] . Hence, another method to find the value of the angles are:
 \[\Rightarrow 2x+3x={{90}^{\circ }}\]
 \[\Rightarrow 5x={{90}^{\circ }}\]
 \[\Rightarrow x={{18}^{\circ }}\]
Hence, the angle are given as:
 \[2x=2\times 18\] and \[3x=3\times 18\] .
 \[36\] and \[54\] .
Therefore, the two acute angles of the triangles are given as \[36\] and \[54\] with the third one as \[{{90}^{\circ }}\] .