Let's solve the problem step by step.

Given Information:

• $A, B, C, D$ are points on the circumference of a circle with center $O$.
• $\angle A = 55^\circ$.
• $x$ and $y$ are angles to be determined.

(a) Find $x$:

From the diagram, we observe that $x$ is an angle at the center subtended by arc $BC$, while $\angle BAC$ is an angle at the circumference subtended by the same arc.

Using the angle at the center theorem: $\text{Angle at the center} = 2 \times \text{Angle at the circumference}$ $x = 2 \times 55^\circ = 110^\circ$

Thus, the correct answer is $x = 110^\circ$.

(b) Find $y$:

From the diagram, $y$ is an angle in the isosceles triangle $ODC$, where $OC = OD$ as radii of the circle.

Since the total sum of angles in a triangle is $180^\circ$, we use: $\angle COD = x = 110^\circ$ $\angle ODC = \angle OCD = y$ $\text{Sum of angles in } \triangle ODC: 110^\circ + 2y = 180^\circ$ $2y = 70^\circ$ $y = 35^\circ$

Thus, the correct answer is $y = 35^\circ$.

Final Answers:

(a) $x = 110^\circ$
(b) $y = 35^\circ$

Would you like a more detailed explanation or another example?

Here are five related questions to expand your understanding:

1. What is the angle at the center theorem, and why does it work?
2. How do we identify isosceles triangles in circle geometry problems?
3. What other circle theorems could be applied in this problem?
4. How would the answer change if the given angle at $A$ was different?
5. Can this method be used for any quadrilateral inscribed in a circle?

Tip: Always look for radii forming isosceles triangles when solving circle problems!