Question

In a triangle other than an equilateral triangle, angle opposite the longest side is greater than

• A. a right angle
• B. $\frac{4}{3}$ of a right angle
• C. $\frac{5}{3}$ of a right angle
• D. $\frac{2}{3}$ of a right angle

Answer 1

The correct option is D

$\frac{2}{3}$ of a right angle

Explanation of the correct option:

Consider $△PQR$ where $PR$ is the longest side.
So we get $PR>PQ$
$\Rightarrow$ $\angle Q>\angle R$ …………....(1)
We also know that $PR>QR$
$\Rightarrow$ $\angle Q>\angle P$ …………....(2)
By adding both the equations,
$\angle Q+\angle Q>\angle R+\angle P$
So we get,
$2\angle Q>\angle R+\angle P$
By adding $\angle Q$ on both L.H.S. and R.H.S.,
$2\angle Q+\angle Q>\angle R+\angle P+\angle Q$
We know that $\angle R+\angle P+\angle Q=180°$ ( Using Angle Sum Property)
So we get,
$3\angle Q>180°$
By division
$\angle Q>60°$
So we get
$\angle Q>\frac{2}{3}(90°)$ ($\because 60°=\frac{2}{3}$of right angle)
i.e. $\angle Q>\frac{2}{3}$ of a right angle
Therefore, it is proved that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater than $\frac{2}{3}$ of a right angle.

Hence option $D$ is the correct answer.