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!