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The angles of a triangle are in the ratio of 3:4:5. Find the smallest angle.

Last updated date: 02nd Aug 2024
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Hint: Firstly, we will assume the ratio to be a variable. Then we will assume the angles in terms of variables. Then we will add the assumed angles and equate them with ${180}^0$. We will value the ratio and hence value of angles. Then, we will determine the smallest angle among them.

Complete step by step solution:
Let’s the angles of the triangle $ABC$ be in the ratio $3:4:5$
And let the ratio to be $x$
So, Angle $A = 3x$ , Angle $B = 4x$ , and Angle $C = 5x$.
We know that the sum of the interior angles of a triangle is ${180}^0$
So, Angle $A$ + Angle $B$+ Angle $C$= ${180}^0$
$3x + 4x + 5x = {180^0} \\$
$\Rightarrow 12x = {180^0} \\$
$\Rightarrow x = {15^0} \\$
Hence, the value of angles
$A = 3x = 3({15^0}) = {45^0} \\$
$B = 4x = 4({15^0}) = {60^0} \\$
$C = 5x = 5({15^0}) = {75^0} \\$
Therefore, the value of the smallest angle is ${45^0}$.

Note:
A triangle has 3 sides and 3 angles. The same question can be asked for a quadrilateral which has 4 sides and 4 angles and whose Sum of interior angles is ${360}^0$.