Question

Two angles of a triangle are in the ratio $4:5$. If the sum of these angles is equal to the third angle, find the angles of the triangle.

Answer 1

Let the common multiple of all the three angles be $x$.Then, the first angle will be $4x$. And the second angle will be $5x$.
In a triangle, sum of all the three angles will be equal to ${180}^{\circ }$
Thus, the third angle $={180}^{\circ }-(4x+5x)$, where $4x+5x$ is sum of first two angles.
$={180}^{\circ }-9x$.......(1)
Given third angle $=4x+5x$ ....(2)
Thus, equation (1) equals to equation (2)
So, $4x+5x={180}^{\circ }-9x$
$\Rightarrow 9x={180}^{\circ }-9x$
$\Rightarrow 9x+9x={180}^{\circ }$
$\Rightarrow 18x={180}^{\circ }$
$\Rightarrow x=\frac{{180}^{\circ }}{18}$
$\therefore x={10}^{\circ }$
Hence, First angle $=4x=4\times 10={40}^{\circ }$,
Second angle $=5x=5\times 10={50}^{\circ }$ and
Third angle $=4x+5x=9x=9\times 10={90}^{\circ }$