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Let's analyze both parts of the problem step by step:

### Part (i):
1. The angle outside the circle is given as \(135^\circ\), and this angle is subtended by the arc where angle \(b\) is located inside the circle.
2. The sum of angles on a straight line is \(180^\circ\), so the angle at the center of the circle (angle \(b\)) is given by:
   \[
   b = 180^\circ - 135^\circ = 45^\circ
   \]
3. Now, we can use the fact that the triangle inside the circle is an isosceles triangle because two of its sides are radii of the circle. The sum of angles in a triangle is \(180^\circ\), so:
   \[
   a + a + b = 180^\circ
   \]
   Substituting the value of \(b = 45^\circ\):
   \[
   a + a + 45^\circ = 180^\circ
   \]
   Simplifying:
   \[
   2a = 135^\circ
   \]
   \[
   a = 67.5^\circ
   \]
   
Thus, in part (i):
- \( a = 67.5^\circ \)
- \( b = 45^\circ \)

### Part (ii):
1. The angle at the center of the circle is given as \(60^\circ\).
2. In this case, angle \(b\) subtends an arc, and angle \(a\) is an inscribed angle subtended by the same arc. We can use the relationship that the angle subtended at the center is twice the angle subtended on the circle:
   \[
   a = \frac{60^\circ}{2} = 30^\circ
   \]
3. Since the triangle inside the circle is isosceles (with two radii as sides), the angles at the base (where \(b\) is located) must be equal. Using the sum of angles in a triangle:
   \[
   a + b + b = 180^\circ
   \]
   Substituting \(a = 30^\circ\):
   \[
   30^\circ + 2b = 180^\circ
   \]
   Simplifying:
   \[
   2b = 150^\circ
   \]
   \[
   b = 75^\circ
   \]

Thus, in part (ii):
- \( a = 30^\circ \)
- \( b = 75^\circ \)

### Final Answers:
- **Part (i):** \( a = 67.5^\circ \), \( b = 45^\circ \)
- **Part (ii):** \( a = 30^\circ \), \( b = 75^\circ \)

Would you like more details on any step, or have any questions?

Here are some related questions for further understanding:
1. How do we derive the relationship between inscribed and central angles in a circle?
2. What is the general rule for the angles in an isosceles triangle?
3. Can the same approach be applied if the circle is not centered at \(O\)?
4. How does the angle subtended by a diameter relate to a right angle in a circle?
5. What happens to the angle relationships if we move to a different segment of the circle?

**Tip:** In problems involving circles, always look for isosceles triangles formed by radii and use angle-sum properties!

Find the value of a and b in the given circle diagrams.

Learn how to find the values of angles a and b in circle geometry problems using key concepts like the Inscribed Angle Theorem and Isosceles Triangle properties. Suitable for students in Grades 9-10, with step-by-step solutions provided.

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