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# In a right angled triangle, two acute angles are in the ratio 2:3. Find the angles.

Last updated date: 20th Jun 2024
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Hint: Here, we need to find the measure of two acute angles. To solve the question, we will assume the two acute angles to be $2x$ and $3x$ respectively. We will apply the angle sum property to form an equation in terms of $x$. We will solve the equation to find the value of $x$ and then using its value we will find the measure of the two acute angles.

We will use the angle sum property of a triangle to find the measure of the angles of the triangle.

It is given that the two acute angles are in the ratio $2:3$.
Let the two acute angles be $2x$ and $3x$ respectively.
As we know, the third angle of the triangle is a right angle.
Thus, the measure of the third angle is $90^\circ$.
Now, the angle sum property of a triangle states that the sum of the measures of the three interior angles of a triangle is always $180^\circ$.
Thus, the sum of the two acute angles and the right angle will be equal to $180^\circ$.
Therefore, we get
$2x + 3x + 90^\circ = 180^\circ$
We will now solve the equation to find the value of $x$.
Subtracting $90^\circ$ from both sides of the equation, we get
$\begin{array}{l} \Rightarrow 2x + 3x + 90^\circ - 90^\circ = 180^\circ - 90^\circ \\ \Rightarrow 2x + 3x = 90^\circ \end{array}$
Adding the like terms in the equation, we get
$\Rightarrow 5x = 90^\circ$
Dividing both sides by 5, we get
$\begin{array}{l} \Rightarrow \dfrac{{5x}}{5} = \dfrac{{90^\circ }}{5}\\ \Rightarrow x = 18^\circ \end{array}$
Therefore, we get the value of $x$ as $18^\circ$.
Finally, we will substitute the value of $x$ to find the measures of the two acute angles.
Substituting $x = 18^\circ$ in $2x$, we get the first acute angle as
$2x = 2 \times 18^\circ = 36^\circ$
Substituting $x = 18^\circ$ in $3x$, we get the second acute angle as
$3x = 3 \times 18^\circ = 54^\circ$
$\therefore$ The measure of the two acute angles of the right angled triangle are $36^\circ$ and $54^\circ$ respectively.

Note: It is given in the question that the triangle is right angled, which means one of the angles of the triangle is $90^\circ$. If it was given an equilateral triangle, it means that every angle of the triangle is $60^\circ$. We need to also keep in mind that the sum of interior angles of a triangle is $180^\circ$ and not $360^\circ$ which is the sum of interior angles of a quadrilateral.