We are given a quadrilateral with one interior angle of $226^\circ$ and two exterior angles of $35^\circ$ and $42^\circ$. We are asked to find the size of the remaining exterior angle $f$.

1. Problem Description

We are given a quadrilateral with one interior angle of $226^\circ$ and two exterior angles of $35^\circ$ and $42^\circ$. We are asked to find the size of the remaining exterior angle $f$.

2. Solution Steps

Let's denote the interior angles of the quadrilateral as $A, B, C, D$.

We are given that one of the interior angles is $226^\circ$. Let $B = 226^\circ$.

We are given two exterior angles: $35^\circ$ and $42^\circ$.

The exterior angle and the interior angle at a vertex add up to $180^\circ$.

Let the interior angle at one vertex be $A$, and the exterior angle at that vertex be $35^\circ$. Then

$A + 35^\circ = 180^\circ$

$A = 180^\circ - 35^\circ = 145^\circ$

Let the interior angle at another vertex be $C$, and the exterior angle at that vertex be $42^\circ$. Then

$C + 42^\circ = 180^\circ$

$C = 180^\circ - 42^\circ = 138^\circ$

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

$A + B + C + D = 360^\circ$

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

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

$D = 360^\circ - 509^\circ = -149^\circ$

There seems to be something wrong. It should have been a reflex angle of $226^\circ$ instead of an interior angle of $226^\circ$. We can form a quadrilateral by extending the sides.

Let the interior angles be $a, b, c$. The external angles are $35^\circ, 42^\circ$, and $f$.

Therefore, $a = 180^\circ - 35^\circ = 145^\circ$, $c = 180^\circ - 42^\circ = 138^\circ$.

The interior angle at $226^\circ$ is $360^\circ - 226^\circ = 134^\circ$. $b = 134^\circ$.

The sum of the angles in a triangle is $180^\circ$.

We have $a + b + c + f + 35^\circ + 42^\circ = 360^\circ$, where $a, b, c$ are interior angles, and $35^\circ, 42^\circ, f$ are external angles of a quadrilateral.

Another interpretation of the figure:

The figure looks like a triangle with one side extended.

So we can say that the sum of the angles $35^\circ$ and $42^\circ$ equals the exterior angle at the $226^\circ$.

$35^\circ + 42^\circ = 77^\circ$, this is the interior angle at the vertex. Therefore $226^\circ$ is an exterior angle.

Since $f$ is an exterior angle of the triangle, $f + 35^\circ + 42^\circ = 180^\circ + 180^\circ = 360^\circ$ is not correct.

Let the missing angle f be such that the polygon consists of 4 angles. The interior angles of this polygon are $180-35=145, 360-226=134, 180-42=138, F$. The sum of the interior angles of a 4-sided polygon is

3

6

0. Therefore, $145+134+138+F=360, F=360-417=-57$. The polygon must be non-convex.

Let's see. $f=180-(180-35)-(180-42)+(360-226)$.

The exterior angle is $f$. The reflex angle is

2

2

6. $f = 360 - 35 - 42 - 226$. Then $f = 57^\circ$.

The angles in the triangle are then $145$, $138$ and angle f which corresponds to the angle with 226 on the other side.

If the triangle angles add up to $180$, we would have $226 =$ the two non adjacent triangle angles

We know 35 + 42 =77 is one angle.

So f= 180 - 77=103

Then the exterior angle would be $360-103-35-42 = 226$

Let's look for another way.

The exterior angle at the reflex angle vertex is $360^\circ - 226^\circ = 134^\circ$

Since the sum of external angles is always $360^\circ$, we have

$35^\circ + 42^\circ + f = 360^\circ - 134^\circ = 360 - 134$.

Then f + $35^\circ + 42^\circ$ is equal to something. We know 35 + 42= 77 and $180-77=103$

Therefore the angle opposite to $f=77$ is $226$, the interior is $134$ and $145+138+77=360$.

The total external angles is 77+103+90

$f + 35 + 42 =103^\circ$. So the interior is

7

7. Final Answer: The final answer is $\boxed{103}$

3. Final Answer

103