Question

Two adjacent angles are in ratio $5:3$ and they together form an angle of ${{128}^{\circ }}$ , find these angles.

Answer 1

Hint: Here we have been given a ratio between two angles and also the sum of the two angles and we have to find these angles. Firstly we will let the two angles by the use of the ratio given then we will add them and put them equal to the sum given. Finally we will solve the equation and get the value of the unknown variable and find the two angles to get our desired answer.

Complete step-by-step solution:
The ratios between two adjacent angles are given as follows:
$5:3$…..$\left( 1 \right)$
The sum of the two angles is as follows,
${{128}^{\circ }}$…..$\left( 2 \right)$
Let from the equation (1) the two angles are,
$5x,3x$….$\left( 3 \right)$
So the sum of above two angles will be equal to equation (2) which we can write as follows,
$5x+3x={{128}^{\circ }}$
$\Rightarrow 8x={{128}^{\circ }}$
Dividing both side by $8$ we get,
$\Rightarrow \dfrac{8x}{8}=\dfrac{{{128}^{\circ }}}{8}$
$\Rightarrow x={{16}^{\circ }}$
Put the above value in equation (3) we get,
$5\times {{16}^{\circ }},3\times {{16}^{\circ }}$
${{80}^{\circ }},{{48}^{\circ }}$
Hence the two angles whose ratio is $5:3$ and sum is ${{128}^{\circ }}$ is ${{80}^{\circ }},{{48}^{\circ }}$.

Note: Angles are formed when two rays of line meet at a common end-point where the line is known as the side of the angle and the common end-point is known as the vertex of the angles. Adjacent angles are those which have a common vertex and a common side but they don’t overlap. The ratios are used by taking the number in it with an unknown variable multiplied to them so that our calculation becomes easy. We can cross check our answer by checking whether the angles obtained satisfy the condition given. In this case we get the angles as ${{80}^{\circ }},{{48}^{\circ }}$ so there ratio is as follows,
$\Rightarrow \dfrac{{{80}^{\circ }}}{{{48}^{\circ }}}$
Divide both numerator and denominator by ${{16}^{\circ }}$ as follows,
$\Rightarrow \dfrac{5}{3}$
Also ${{80}^{\circ }}+{{48}^{\circ }}={{128}^{\circ }}$
Hence our answer is correct.