Question

An exterior angle of a triangle is $105°$ and its two interior opposite angles are equal. Each of these equal angles is

• A. $37\frac{1}{2}°$
• B. $52\frac{1}{2}°$
• C. $72\frac{1}{2}°$
• D. $75°$

Answer 1

The correct option is B

$52\frac{1}{2}°$

Exterior Angle Theorem.

Explanation for the correct option:

Option (B). $52\frac{1}{2}°$

Exterior Angle Theorem.

When one of the sides of a triangle is extended, the external angle formed between the extended side and its adjacent side is called an exterior angle.

The Exterior Angle Theorem states that “The exterior angle is equal to the sum of interior opposite angles of a triangle”.

Consider the $∆\mathrm{ABC}$. Draw an exterior angle to the $∆\mathrm{ABC}$ by extending the side $\mathrm{BC}$.

Given the exterior angle of a triangle is $105°$ and its two interior opposite angles are equal.

Therefore, $\angle \mathrm{ACD}=105°$ and let $\angle \mathrm{BAC}=\angle \mathrm{ABC}=\alpha$.

Hence, by the Exterior Angle Theorem,

$\begin{array}{rcl}\angle \mathrm{ACD}& =& \angle \mathrm{BAC}+\angle \mathrm{ABC}\\ & \Rightarrow & 105°=\alpha +\alpha \\ & \Rightarrow & 105°=2\alpha \\ & \Rightarrow & \alpha =\frac{105°}{2}\\ & \Rightarrow & \alpha =52\frac{1}{2}°\end{array}$

Therefore, each of the interior opposite angles is equal to $52\frac{1}{2}°$.

Hence, the correct option is (B) i.e., $52\frac{1}{2}°$