1. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

2. O is the circumcenter of triangle ABC. If \(\angle\)BAC = 85° and \(\angle\)BCA = 70°, then what is the measure of \(\angle\)OAC?

(a) \(65^o\) (b) \(42\cfrac{1}{2}^o\) (c) \(50^o\) (d) \(25^o\)

3. In triangle ABC, the circumcenter is O. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)OAC?

(a) 70° (b) 40° (c) 110° (d) 140°

4. In triangle ABC, O is the circumcenter and D is the midpoint of side BC. If \(\angle\)BAC = 40°, find the measure of \(\angle\)BOD.

5. If O is the circumcenter of triangle ABC and \(\angle\)OAB = 50°, then the measure of \(\angle\)ACB is —.

6. In triangle ABC, the circumcenter is \(O\). If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)AOC?

7. In triangle ABC, the circumcenter is \(O\) and \(\angle\)OAB = 50°. Determine the value of \(\angle\)ACB.

8. If \(O\) is the circumcenter of \(\triangle ABC\) and \(\angle BAC = 50^\circ\), then find the measure of \(\angle OBC\).

(a) 100° (b) 30° (c) 40° (d) 50°

9. \(O\) is the circumcenter of triangle \(ABC\). If \(\angle ABC = 65^\circ\) and \(\angle BCA = 40^\circ\), then the measure of \(\angle BOC\) is _____?

10. In the isosceles triangle ABC, AB = AC. The circumcenter of the triangle is O, and it lies on the side of BC opposite to the vertex A. If \(\angle\)BOC = 100°, calculate and write the measures of \(\angle\)ABC and \(\angle\)ABO.

11. In the adjacent figure, O is the circumcenter of triangle ΔABC and \(\angle\)AOC = 110°; calculate and write the measure of \(\angle\)ABC.

12. In triangle ABC, O is the circumcenter and D is the midpoint of side BC. If \(\angle\)BAC = 40°, find the measure of \(\angle\)BOD.

13. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

14. In triangle ABC, the circumcenter is O. It is given that \(\angle\)BAC = 85° and \(\angle\)BCA = 55°. Find the value of \(\angle\)OAC.

15. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

16. In triangle \(\triangle ABC\), O is the circumcenter. If \(\angle OAB = 35^\circ\), then what is the measure of \(\angle ACB\)?

(a) 45° (b) 50° (c) 55° (d) 60°

17. O is the circumcenter of triangle ABC; if \(\angle\)BOC = 80°, then the measure of \(\angle\)BAC is —.

(a) 40° (b) 160° (c) 130° (d) 110°

18. In triangle \(\triangle ABC\), the incenter is \(O\), and \(\angle BOC = 120^\circ\). Find the measure of \(\angle BAC\).

19. In a circle with center O, triangle ABC is inscribed. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then find the measure of \(\angle\)AOC.

20. O is the circumcenter of triangle ABC. If \(\angle\)OAB = 50°, then the measure of \(\angle\)ACB is 40°.

21. In triangle ABC, if O is the orthocenter and ∠BOC = 120°, then what is the measure of ∠BAC?

(a) 80° (b) 60° (c) 90° (d) 75°

22. In triangle ABC, the incenter is O, and \(\angle\)BOC = 120°. What is the measure of \(\angle\)BAC?

23. If \( O \) is the circumcenter of triangle \( ABC \) and \( \angle BAC = 40^\circ \), determine the value of \( \angle OBC \).

(a) 100° (b) 30° (c) 50° (d) 40°

24. \(O\) is the circumcenter of \(\triangle ABC\). If \(\angle OAB = 50^\circ\), then find the measure of \(\angle ACB\).

(a) 50° (b) 100° (c) 80° (d) 40°

25. In \(\triangle\)ABC, the circumcenter is \(O\) and \(\angle\)BAC = \(65^\circ\). Determine the value of \(\angle\)OBC.

26. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____.

(a) 30° (b) 45° (c) 60° (d) 90°

27. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____.

(a) 30° (b) 45° (c) 60° (d) 90°

28. An equilateral triangle where each side is 6 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

29. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

30. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]