The question asks for the measure of the exterior angle at the right-angled vertex of a right triangle, given that one acute angle is $30^\circ$. Let's break this down:

Right Triangle Angle Properties

• A right triangle is defined as a triangle containing one angle that measures exactly $90^\circ$. This is the right-angled vertex.
• The sum of the interior angles in any triangle is always $180^\circ$.
• In a right triangle, the sum of the two acute angles must be $90^\circ$ ($180^\circ - 90^\circ = 90^\circ$).

Exterior Angle Calculation

1. Identify the interior angle at the right-angled vertex: By definition, this is $90^\circ$.
2. Use the supplementary angle property: An exterior angle and its adjacent interior angle form a linear pair, meaning they add up to $180^\circ$.
3. Calculate the exterior angle: Exterior Angle = $180^\circ$ - Interior Angle at the right-angled vertex Exterior Angle = $180^\circ - 90^\circ$ Exterior Angle = $90^\circ$

The fact that one acute angle is $30^\circ$ (implying the other is $60^\circ$) is extra information not needed to find the exterior angle at the $90^\circ$ vertex itself.