In this problem, you are asked to find the measures of two angles: $\angle AOB$ and $\angle COD$ from the given diagram and information.

Given:

1. $2b - a = 65^\circ$
2. $\angle BOC = 90^\circ$
3. $a$ and $b$ are unknown angles.

We'll solve each part step by step.

(i) Find $\angle AOB$:

• From the figure, $\angle AOB = a + b$ (because both $\angle AOB$ is composed of angle parts labeled $a$ and $b$).

• We are given $2b - a = 65^\circ$. We also know that $\angle BOC = 90^\circ$, and it is composed of $2b$, so $2b = 90^\circ$.

  Solving for $b$: $2b = 90^\circ \quad \Rightarrow \quad b = \frac{90^\circ}{2} = 45^\circ$

  Now substitute $b = 45^\circ$ into $2b - a = 65^\circ$: $2(45^\circ) - a = 65^\circ \quad \Rightarrow \quad 90^\circ - a = 65^\circ$ $a = 90^\circ - 65^\circ = 25^\circ$

  Now, using $a = 25^\circ$ and $b = 45^\circ$: $\angle AOB = a + b = 25^\circ + 45^\circ = 70^\circ$

Thus, $\angle AOB = 70^\circ$.

(ii) Find $\angle COD$:

• From the diagram, $\angle COD = 2a + b + 15^\circ$.

  Substituting the values of $a = 25^\circ$ and $b = 45^\circ$: $\angle COD = 2(25^\circ) + 45^\circ + 15^\circ = 50^\circ + 45^\circ + 15^\circ = 110^\circ$

Thus, $\angle COD = 110^\circ$.

Final Answers:

1. $\angle AOB = 70^\circ$
2. $\angle COD = 110^\circ$

Would you like any additional details or clarification?

Here are five related questions:

1. How do we handle angle subtraction and addition in complex geometries?
2. What is the significance of vertical and adjacent angles in geometry?
3. Can angles in polygons be calculated similarly to the ones in this problem?
4. How can systems of equations simplify angle problems in geometry?
5. What are other common angle relationships used in geometric proofs?

Tip: When solving for unknowns in geometric figures, setting up and solving simultaneous equations is often key.