Question

In a $\vartriangle ABC$ , if $2\angle A = 3\angle B = 6\angle C$ , calculate $\angle A$ , $\angle B$ and $\angle C$ .

Answer 1

In the above given question, we are given a triangle $\vartriangle ABC$ in which the three angles $\angle A$ , $\angle B$ and $\angle C$ are given as such that two times of $\angle A$ is equal to three times of $\angle B$ which is then equal to six times of $\angle C$ . We have to calculate the values of the measure of all the three angles $\angle A$ , $\angle B$ and $\angle C$ . In order to approach the required solution, we have to consider the property of a triangle known as the angle sum property of a triangle.

Complete answer:
Given that, a triangle $\vartriangle ABC$ such that,
$\Rightarrow 2\angle A = 3\angle B = 6\angle C$
We have to find the value of the three angles $\angle A$ , $\angle B$ and $\angle C$ .
Now since it is given that $2\angle A = 3\angle B = 6\angle C$
Hence, we can also write is as,
$\Rightarrow 2\angle A = 6\angle C$
That is,
$\Rightarrow \angle A = 3\angle C$ ...(1)
Similarly, we can write $2\angle A = 3\angle B = 6\angle C$ as,
$\Rightarrow 3\angle B = 6\angle C$
That is,
$\Rightarrow \angle B = 2\angle C$ ...(2)
Now, since we know that from the angle sum property of the triangle, the sum of all three angles in a triangle is equal to $180^\circ$ .
Therefore, applying the angle sum property of a triangle in the given triangle $\vartriangle ABC$ , we can write
$\Rightarrow \angle A + \angle B + \angle C = 180^\circ$
Substituting the values from equations 1 and 2 in this equation, we get
$\Rightarrow 3\angle C + 2\angle C + \angle C = 180^\circ$
That gives us,
$\Rightarrow 6\angle C = 180^\circ$
Hence,
$\Rightarrow \angle C = 30^\circ$
Therefore,
$\Rightarrow \angle A = 3\angle C = 3 \times 30^\circ$
That is,
$\Rightarrow \angle A = 90^\circ$
And,
$\Rightarrow \angle B = 2\angle C = 2 \times 30^\circ$
That is,
$\Rightarrow \angle B = 60^\circ$
Therefore, the three angles of $\vartriangle ABC$ are $\angle A = 90^\circ$ , $\angle B = 60^\circ$ and $\angle C = 30^\circ$ .

Note:
We can note that in the above given triangle $\vartriangle ABC$ , we have $\angle A = 90^\circ$ . Now, an angle of $\vartriangle ABC$ is a right angle, therefore the above given triangle $\vartriangle ABC$ is a right angled triangle. Hence, the side opposite to the right angle $\angle A = 90^\circ$ , that is the side BC, is the hypotenuse for the triangle $\vartriangle ABC$ while the other two sides AB and AC are the perpendicular sides.