The problem asks us to find the size of angle $f$ in the given figure. The figure is a quadrilateral with one interior angle of $226^\circ$ and two exterior angles of $35^\circ$ and $42^\circ$ respectively.

1. Problem Description

The problem asks us to find the size of angle $f$ in the given figure. The figure is a quadrilateral with one interior angle of $226^\circ$ and two exterior angles of $35^\circ$ and $42^\circ$ respectively.

2. Solution Steps

First, we need to find the interior angles of the quadrilateral corresponding to the exterior angles of $35^\circ$ and $42^\circ$.

The sum of an interior angle and its corresponding exterior angle is $180^\circ$.

So, the interior angle corresponding to the $35^\circ$ exterior angle is $180^\circ - 35^\circ = 145^\circ$.

The interior angle corresponding to the $42^\circ$ exterior angle is $180^\circ - 42^\circ = 138^\circ$.

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

Let $f$ be the unknown angle. Then, we have:

$f + 145^\circ + 138^\circ + 226^\circ = 360^\circ$

$f + 509^\circ = 360^\circ$

However, this doesn't make sense because it would mean $f$ is negative. Instead, the figure is a concave quadrilateral. Consider the interior reflex angle to be $360^\circ - 226^\circ = 134^\circ$.

Then, we have:

$f + 145^\circ + 138^\circ + 134^\circ = 360^\circ$

$f + 417^\circ = 360^\circ$. Again, this doesn't make sense.

The angle of $226^\circ$ is an external angle. Hence, consider a triangle formed by extending the lines.

The internal angle at the bottom left is $35^\circ$, and the internal angle at the bottom right is $42^\circ$.

The angle adjacent to $226^\circ$ is $360^\circ - 226^\circ = 134^\circ$.

The sum of angles in a quadrilateral is $360^\circ$.

Thus, $35^\circ + 42^\circ + x + f = 360^\circ$, where x is the supplementary angle of

2

2

6. $x = 360 - 226 = 134$

So, $f + 35 + 42 = 226$ because the exterior angle of the vertex is equal to sum of the angles at the remote interior vertices.

$f + 77 = 226$

$f = 226 - 77$

$f = 149$

3. Final Answer

The final answer is $\boxed{149}$