Question

One of the exterior angle of a triangle is ${105}^0$ and the interior opposite angles are in the ratio 2 : 5 . Find the angles of the triangle.

• A. ${30}^o;{45}^o;{105}^o$
• B. ${45}^o;{45}^o;{90}^o$
  
• C. ${30}^o;{75}^o;{75}^o$
  
• D. ${60}^o;{30}^o;{90}^o$
  

Answer 1

Answer: (C)

${30}^o;{75}^o;{75}^o$

We have the property that, in a triangle exterior angle is equal to the sum of interior opposite angles.

Given, the interior opposite angles to the exterior angle ${105}^{\circ }$ are in the ration $2:5$

$\therefore 2x+5x={105}^o$

$7x={105}^o$ $⟹x={15}^o$

Therefore the interior opposite angles to the angle ${105}^o$ are $2x=2\left(15\right)={30}^o$ and $5x=5\left(15\right)={75}^o$

Let the third angle of the triangle be $C$

We have the sum of interior angles of a triangle is ${180}^o$

$\therefore {30}^o+{75}^o+C={180}^o$

$C={180}^o-{75}^o-{30}^o={180}^o-{105}^o$

$C={75}^o$

Hence, the angles are ${30}^o,{75}^o,{75}^o$.