We are given a quadrilateral with one interior angle labeled as $f$, and three exterior angles measuring $35^{\circ}$, $42^{\circ}$, and $226^{\circ}$. We need to find the value of the interior angle $f$.

1. Problem Description

We are given a quadrilateral with one interior angle labeled as $f$, and three exterior angles measuring $35^{\circ}$, $42^{\circ}$, and $226^{\circ}$. We need to find the value of the interior angle $f$.

2. Solution Steps

First, we need to find the three interior angles of the quadrilateral that are adjacent to the given exterior angles. The sum of an interior angle and its corresponding exterior angle is $180^{\circ}$.

So,

Interior angle 1 = $180^{\circ} - 35^{\circ} = 145^{\circ}$

Interior angle 2 = $180^{\circ} - 42^{\circ} = 138^{\circ}$

Interior angle 3 = $360^{\circ} - 226^{\circ} = 134^{\circ}$ (Note that an exterior angle greater than 180 is referred to as a reflex angle)

The sum of the interior angles of a quadrilateral is $360^{\circ}$.

Therefore, $f + 145^{\circ} + 138^{\circ} + 134^{\circ} = 360^{\circ}$

$f + 417^{\circ} = 360^{\circ}$

$f = 360^{\circ} - 417^{\circ}$

$f = -57^{\circ}$. This is impossible, since angle f should be positive. Let's go back to step 1: the reflex angle is outside the quadrilateral, not inside, so we made an error.

The correct calculation for interior angle 3 is $180 - (360-226) = 180 - 134 = 46^{\circ}$.

Now we have $f + 145^{\circ} + 138^{\circ} + 46^{\circ} = 360^{\circ}$

$f + 329^{\circ} = 360^{\circ}$

$f = 360^{\circ} - 329^{\circ}$

$f = 31^{\circ}$

3. Final Answer

$f = 31^{\circ}$