Question 1: In a circle, two angles subtend the same arc. If one angle measures 60°, what is the measure of the other angle?

Answer: 60

Question 2: Construct two angles on a circle that subtend the same arc. If one angle is 45°, what should the measure of the second angle be?

Answer: 45

Question 3: A student claims that angles subtended by the same arc can have different measures. Is this true or false? Explain your answer.

Answer: False. Angles subtended by the same arc are always equal.

Question 4: Identify the mistake in the following reasoning: 'Angles subtended by the same arc are not necessarily equal because they can be on different sides of the circle.'

Answer: The mistake is assuming angles on different sides of the circle can be equal; angles subtended by the same arc are only equal when on the same side.

Question 5: If two angles in a circle are both 70° and subtend the same arc, what can you conclude about the points where these angles are formed?

Answer: They are on the same side of the circle, subtending the same arc.

Question 6: Calculate the angles formed at a point outside a circle when two secants intersect, given that the intercepted arcs are 80° and 120°.

Answer: 80 + 120 = 200; 200/2 = 100°

Question 7: A diagram shows two angles, both 50°, subtending the same arc. The student claims they are not equal because one is an inscribed angle and the other is an angle at the center. Is the student correct? Why or why not?

Answer: No, because if both angles subtend the same arc, they are equal regardless of their position.

Question 8: An error is made in a diagram where an angle subtended by the same arc is measured as 75° on one side of the circle and 60° on the other. Identify the mistake and correct it.

Answer: Angles subtended by the same arc on the same side must be equal; the 60° measurement is incorrect.

Question 9: Prove that angles subtended by the same arc are equal using circle theorems and reasoning.

Answer: Angles subtended by the same arc are inscribed angles sharing the same intercepted arc; inscribed angles subtend equal arcs, so the angles are equal.

Question 10: In a circle, two angles of 80° and 70° are shown. One is inscribed, and the other is at the circumference but on different sides of the same arc. Is this possible? Explain.

Answer: No, because only angles on the same side of the circle subtending the same arc are equal; they cannot differ like this.

Question 11: A challenge problem: Two angles, both 65°, are inscribed in a circle, subtending the same arc. One is at the end of a diameter, and the other is somewhere else on the circle. Are both measurements correct? Why?

Answer: Yes, both are correct because angles inscribed in the same arc are equal, regardless of their position.