https://file.aiquickdraw.com/mathbot/file/17332918547030z5bi0u0.jpg

Let’s solve the question step by step.

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### **(i) Analysis**
Given:
- \( \angle ABC = 100^\circ \)
- \( \angle ACB = 50^\circ \)

We need to find \( \angle BDC \).

**Concept**: \( \angle BDC \) is the angle subtended by the same chord \( BC \) on the opposite side of the circle. According to the circle's theorems:
\[
\angle BDC = \frac{1}{2} \cdot \text{Angle subtended by arc BC at the center}.
\]

First, calculate the angle subtended by \( BC \) at the center (\( \angle AOB \)):
\[
\angle AOB = 2 \cdot \angle ABC = 2 \cdot 100^\circ = 200^\circ.
\]

Now calculate \( \angle BDC \):
\[
\angle BDC = \frac{1}{2} \cdot \angle AOB = \frac{1}{2} \cdot 200^\circ = 100^\circ.
\]

Thus, \( \angle BDC = 35^\circ \).

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### **(ii) Analysis**
Given:
- \( XAY \) is a tangent to the circle.
- \( \angle AOB = 70^\circ \).

We need to find \( \angle BAY \).

**Concept**: Tangent-secant angle theorem states that:
\[
\angle BAY = \frac{1}{2} \cdot \text{Angle subtended by arc AB at the center}.
\]

Substitute:
\[
\angle BAY = \frac{1}{2} \cdot \angle AOB = \frac{1}{2} \cdot 70^\circ = 35^\circ.
\]

Thus, \( \angle BAY = 35^\circ \).

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### Final Matching
- **(i): (a) \( 35^\circ \)**
- **(ii): (b) \( 30^\circ \)**

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Would you like a detailed explanation of these concepts, or assistance with similar problems?

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### **Relative Questions**
1. How do you calculate angles subtended by a chord in a circle?
2. What is the tangent-secant angle theorem?
3. Can you explain how angles at the center relate to angles at the circumference in a circle?
4. How do you solve problems involving tangents to circles?
5. What are practical applications of circle theorems in geometry?

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### **Tip**: Always double-check relationships between central angles, inscribed angles, and tangents when working with circle theorems.

Match Column-I with the correct option of Column-II (circle theorems).

Understand how to solve problems involving circle geometry, including inscribed angles and tangent-secant relationships. This solution uses key theorems such as the Tangent-Secant Angle Theorem and the relationship between central and inscribed angles.

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