Question

An exterior angle is drawn to a triangle; if this exterior angle is acute, then what type of a triangle will be formed?

Answer 1

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 $∆\mathrm{ABC}$.
Since the sum of all the angles of a triangle is equal to $180°$, therefore
$\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180°$

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

Therefore, by the Exterior Angle Theorem, $\angle \mathrm{ACD}=\angle \mathrm{BAC}+\angle \mathrm{ABC}$

Given that this exterior angle is acute i.e.,

$\begin{array}{rcl}\angle \mathrm{ACD}& <& 90°\\ & \Rightarrow & \angle \mathrm{BAC}+\angle \mathrm{ABC}<90°\end{array}$

If the sum of two interior angles of a triangle is less than $90°$, then the remaining angle must be greater than $90°$.

Therefore, $\angle \mathrm{ACB}>90°$ i.e., obtuse.

Since one of the angles of the triangle is greater than $90°$, so $∆\mathrm{ABC}$ will be an obtuse-angled triangle.

Hence, if an exterior angle drawn to a triangle is acute, then the triangle formed will be an obtuse-angled triangle.