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In a triangle ABC if $2\angle A = 3\angle B = 6\angle C$,then find A,B,C .

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Last updated date: 17th Jun 2024
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Answer
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Hint:In this question first let us suppose that $2\angle A = 3\angle B = 6\angle C$ = $x$ . Now try to find out the values of $\angle$ A , $\angle$ B and $\angle $C in the terms of $x$ After that we know that the sum of the interior angles of the triangle is ${180^\circ }$ . By using this property we will find out the value of $x$ and the remaining angles.

Complete step-by-step answer:
From this let us suppose that the t $2\angle A = 3\angle B = 6\angle C$ = $x$
Therefore ;
$2\angle A = x$ , $\angle A = \dfrac{x}{2}$
Similarly ;
$3\angle B = x,\angle B = \dfrac{x}{3}$ and $6\angle C = x,\angle C = \dfrac{x}{6}$
As we know that the sum of the interior angle of a triangle is ${180^\circ }$ .
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That means $\angle A + \angle B + \angle C = {180^\circ }$ ,
Now try to write the angle A,B and C in the terms of $x$ , As above we prove that
$\angle A = \dfrac{x}{2}$ , $\angle B = \dfrac{x}{3}$ and $\angle C = \dfrac{x}{6}$
Therefore
$\dfrac{x}{2} + \dfrac{x}{3} + \dfrac{x}{6} = {180^\circ }$
So the L.C.M of $2,3,6$ is $6$
Hence change the numerator according to this so we get as ;
$\dfrac{{3x + 2x + x}}{6} = {180^\circ }$
Now multiple by $6$ on both side we get ;
$3x + 2x + x = 180 \times 6$
$6x = 180 \times 6$
Hence $x = 180$
As we know that the $\angle A = \dfrac{x}{2}$ , $\angle B = \dfrac{x}{3}$ and $\angle C = \dfrac{x}{6}$
hence
$\angle A = \dfrac{{180}}{2}$ $\angle B = \dfrac{{180}}{3}$ $\angle C = \dfrac{{180}}{6}$
therefore
$\angle A = {90^\circ }$ $\angle B = {60^\circ }$ and $\angle C = {30^\circ }$

Note:Whenever we have found some value of angle, always consider that it is equal to x . Now try to find out some relation between them and use the properties of triangles to proceed further .As in this the angle A is ${90^\circ }$ hence it is a right angle triangle .
Equilateral Triangle : In which all the sides are equal in length . In this triangle all the angles are ${60^ \circ }$.