The two problems in the image involve circles with inscribed triangles and focus on determining the unknown angles labeled as $x^\circ$. Let's address each case:

Part (a)

• Given:

  • $\triangle AOB$ is inscribed in the circle with center $O$.
  • Angle $\angle OBC = 30^\circ$.
  • The angle $x^\circ$ is at point $A$, subtended by arc $BC$.

• Solution: The key here is that $\angle AOC$ is the angle subtended by the same arc $BC$ at the center of the circle. According to the inscribed angle theorem, the angle subtended at the center is twice the angle subtended at the circumference by the same arc.

Thus: $\angle AOC = 2 \times \angle ABC$ Since $\angle OBC = 30^\circ$, we have: $x^\circ = 2 \times 30^\circ = 60^\circ$

Part (b)

• Given:

  • $\triangle PQR$ is inscribed in the circle.
  • Angle $\angle PQR = 50^\circ$.
  • Angle $x^\circ$ is subtended by arc $PR$ at point $O$.

• Solution: Again, using the inscribed angle theorem, the angle subtended at the center $O$ is twice the angle subtended at the circumference by the same arc $PR$.

Thus: $\angle POR = 2 \times \angle PQR$ Given $\angle PQR = 50^\circ$, we get: $x^\circ = 2 \times 50^\circ = 100^\circ$

Summary of answers:

• Part (a): $x = 60^\circ$
• Part (b): $x = 100^\circ$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the inscribed angle theorem?
2. How can you find the angle at the center of a circle subtended by a chord?
3. What are the properties of angles in a cyclic quadrilateral?
4. Can the angles at the circumference ever equal the angles at the center?
5. What is the relationship between an arc and the angles subtended by it?

Tip: In a circle, the angle at the center is always twice the angle at the circumference for the same arc.