Question

The angle of a triangle is in the ratio of \[1:3:5\]. Find the measure of each angle.

Answer 1

Hint: At first, we should know the sum of all angles of a triangle. Then assuming one of the angles of the triangle be x and forming a linear equation, we can solve this problem.

Complete step-by-step answer:
Here the angle of the triangle is the ratio\[1:3:5\].
Let us assume, one of the angles is $x$. And then according to the ratio of angles of triangle,
We have all angles as x, 3x, 5x.
We know the sum of all angles of a triangle is ${180^\circ}$.
So, forming the equation,
     $x + 3x + 5x = {180^\circ}$
Solving it , we get
     $ \Rightarrow 9x = {180^\circ}$
     $ \Rightarrow x = {20^\circ}$
Now, we will substitute the value of x to find the other two angles.
Hence the angle taken as ‘$x$’ is $ \Rightarrow {20^\circ}$
The angle taken as ‘$3x$’ is $ \Rightarrow 3 \times {20^\circ} = {60^\circ}$
The angle taken as ‘$5x$’ is $ \Rightarrow 5 \times {20^\circ} = {100^\circ}$
So, three angles of the triangle are, ${20^\circ}$, ${60^\circ}$ and ${100^\circ}$.

Additional Information: (1) Sum of all angles of triangle is ${180^\circ}$ i.e , if there is a triangle ABC , \[\angle A + \angle B + \angle C = 180^\circ \].
(2) With the given ratio of some terms, we can get their actual values with the help of one single variable only.

Note: An important property of triangle for its three angles is that their sum total is ${180^\circ}$. Also the fundamental rule of ratio and proportions are used in the above problem. We solved this type of problem by assuming one of the angles as $x$ and further applying the different ratio as a triangle and measuring the angle. Linear equations and its solution methods are important here. Calculations should be done very attentively to avoid silly mistakes instead of having key concepts.