To solve the problem in the image, we can break it down step by step.

Part (i): Work out the size of the angle marked $x$.

We are given that:

• $\angle AOB = x$,
• $\angle BOC = 90^\circ$,
• $\angle AOC = 220^\circ$.

Since all three angles, $\angle AOB$, $\angle BOC$, and $\angle AOC$, lie around the point $O$ on a straight line (since $OA$, $OB$, and $OC$ are straight lines), they should add up to $360^\circ$. This is because angles around a point always sum to $360^\circ$.

Thus, we have the equation: $x + 90^\circ + 220^\circ = 360^\circ$ Simplifying: $x + 310^\circ = 360^\circ$ Solving for $x$: $x = 360^\circ - 310^\circ = 50^\circ$

Therefore, $x = 50^\circ$.

Part (ii): Give a reason for your answer.

The reason is that the angles around a point must sum to $360^\circ$. Since $\angle AOB$, $\angle BOC$, and $\angle AOC$ are angles around the point $O$, their sum must equal $360^\circ$. By subtracting the sum of the other two angles ($90^\circ$ and $220^\circ$) from $360^\circ$, we can find the value of $x$.

Do you have any questions or need further clarification? Here are some related questions:

1. What is the rule for the sum of angles on a straight line?
2. How would the problem change if $\angle BOC$ were $100^\circ$ instead of $90^\circ$?
3. Can the sum of angles around a point ever be less than $360^\circ$?
4. What is the relationship between complementary and supplementary angles?
5. How would the calculation change if $\angle AOC$ were $180^\circ$ instead of $220^\circ$?

Tip: Always remember that the sum of angles around a point is $360^\circ$, which is a crucial rule for solving angle problems.