In a triangle $\vartriangle ABC$, if $3\angle A = 4\angle B = 6\angle C$, Calculate the angles.

Hint: In order to solve the question, we have to assume that given data as variables, Then we have to find out the values of the given angle in terms of the assumed variable.

Complete step by step answer:
It is stated in the question that $3\angle A = 4\angle B = 6\angle C$
Now let us consider $3\angle A = 4\angle B = 6\angle C = x$
Therefore we can write-
$3\angle A = x$
$\Rightarrow \angle A = \dfrac{x}{3}.....\left( 1 \right)$
$4\angle B = x$
$\Rightarrow \angle B = \dfrac{x}{4}.....\left( 2 \right)$
$6\angle C = x$
$\Rightarrow \angle C = \dfrac{x}{6}.....\left( 3 \right)$
Since the sum of angles of the triangle is $180_{}^\circ$
Therefore we can write-
$\angle A + \angle B + \angle C = 180_{}^\circ$
Substituting the values of $\angle A,\angle B,\angle C$ from equation $\left( 1 \right)$,$\left( 2 \right)$and $\left( 3 \right)$ we get-
$\Rightarrow \dfrac{x}{3} + \dfrac{x}{4} + \dfrac{x}{6} = 180_{}^\circ$
Now we have to find out the L.C.M of $3,4,6$ and we get-
$\Rightarrow \dfrac{{4x + 3x + 2x}}{{12}} = 180_{}^\circ$
By doing cross multiplication we get-
$\Rightarrow 4x + 3x + 2x = 180_{}^\circ \times 12$
$\Rightarrow 9x = 2160_{}^\circ$
Now by division we get the value of $x$ and we get-
$\Rightarrow x = 240_{}^\circ$
Now by substituting the values of $x$ in equation $\left( 1 \right)$,$\left( 2 \right)$ and $\left( 3 \right)$we will get the values of $\angle A,\angle B,\angle C$.
Therefore the value of $\angle A = \dfrac{{240}}{3} = 80_{}^\circ$
The value of $\angle B = \dfrac{{240_{}^\circ}}{4} = 60_{}^\circ$
The value of $\angle C = \dfrac{{240_{}^\circ}}{6} = 40_{}^\circ$

$\therefore$ The values of $\angle A, \angle B, \angle C$ are $80_{}^\circ,60_{}^\circ,40_{}^\circ$.

Note:
- A triangle is a polygon which has three sides, three vertices and three edges.
- Sum of the angles of the triangle is $180_{}^\circ$
- There are three types of triangle on the basis of their sides that is scalene triangle, isosceles triangle and scalene triangle.
- On the basis of their angle there are three types of triangle that is acute angled triangle, obtuse-angled triangle and right-angled triangle.
- The sum of its three sides is the perimeter of a triangle.
- Exterior angle of a triangle is always equal to the sum of interior angles of a triangle.
- The sum of two sides of a triangle is always more than the other remaining side.