We are given a triangle with one angle labeled as $57^\circ$, and an exterior angle labeled as $116^\circ$. We are asked to find the value of the angle $r$.

1. Problem Description

We are given a triangle with one angle labeled as $57^\circ$, and an exterior angle labeled as $116^\circ$. We are asked to find the value of the angle $r$.

2. Solution Steps

First, we need to find the interior angle adjacent to the exterior angle of $116^\circ$. We know that the sum of an exterior angle and its adjacent interior angle is $180^\circ$.

$Interior \ angle + Exterior \ angle = 180^\circ$

$Interior \ angle + 116^\circ = 180^\circ$

$Interior \ angle = 180^\circ - 116^\circ = 64^\circ$

Now we know two angles of the triangle: $57^\circ$ and $64^\circ$. The sum of the angles in a triangle is $180^\circ$.

$r + 57^\circ + 64^\circ = 180^\circ$

$r + 121^\circ = 180^\circ$

$r = 180^\circ - 121^\circ = 59^\circ$

3. Final Answer

The value of angle $r$ is $59^\circ$.