Question

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

Answer 1

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 ${360}^0$.