Question

An exterior angle of a triangle is \[{{105}^{\circ }}\] and its two interior opposite angles are equal. Each of these equal angles is
\[\begin{align}
  & A.37{{\dfrac{1}{2}}^{\circ }} \\
 & B.52{{\dfrac{1}{2}}^{\circ }} \\
 & C.72{{\dfrac{1}{2}}^{\circ }} \\
 & D{{.75}^{\circ }} \\
\end{align}\]

Answer 1

Hint: In the above question, we will use the property of the triangle that the exterior angle of the triangle is equal to the sum of its opposite interior angle. Exterior angle is already known, so we will suppose the interior angle as x. Then, it is given that both interior angles are equal. So, we will use the above property to find the required angle through the equation formed.

Complete step-by-step answer:
We have been given that an exterior angle of a triangle is \[{{105}^{\circ }}\] and its two interior opposite angles are equal. So, we have to find these equal angles.
We know the property of a triangle that, the exterior angle of a triangle is equal to the sum of its opposite interior angle, as shown in the figure below:

Here, \[\text{In }\Delta \text{ABC, }\angle \text{ACO=}\angle \text{B+}\angle \text{A}\]
We have an exterior angle equal to \[{{105}^{\circ }}\] and both interior angles are equal.
Let the interior angle be x.
\[\begin{align}
  & \Rightarrow x+x={{105}^{\circ }} \\
 & \Rightarrow 2x={{105}^{\circ }} \\
\end{align}\]
On dividing the equation by 2, we get:
\[\begin{align}
  & \Rightarrow \dfrac{2x}{2}=\dfrac{{{105}^{\circ }}}{2} \\
 & \Rightarrow x=52{{\dfrac{1}{2}}^{\circ }} \\
\end{align}\]
Hence, each of the equal angle is \[52{{\dfrac{1}{2}}^{\circ }}\]
Therefore, the correct option is B.

Note: If you do not remember the property in case you can solve it by another method in which first of all you just find the third angle of the triangle using the linear pair property by subtracting \[{{105}^{\circ }}\] from \[{{180}^{\circ }}\]. Then, use the property of a triangle i.e. the sum of internal angles of a triangle is equal to \[{{180}^{\circ }}\] to find the value of x.