Question 1: In a circle, two angles subtend the same arc. If one angle measures 50°, what is the measure of the other angle?

Answer: 50

Question 2: Construct a triangle on the grid so that two angles subtend the same arc and measure 70° and 40°. Label all points clearly.

Answer: Students will draw and label the diagram; no numerical answer.

Question 3: Explain why angles subtended by the same arc are equal. Use a diagram if necessary.

Answer: Angles subtended by the same arc are equal because they intercept the same segment of the circle, and the measure of an inscribed angle depends only on the arc it subtends.

Question 4: A circle has an inscribed angle of 35° subtending an arc. Find the measure of the angle inscribed on the same arc on the opposite side.

Answer: 35

Question 5: Calculate the measure of an angle at the circumference if it subtends an arc of 120° in the circle.

Answer: 60

Question 6: Two angles, 65° and ?°, subtend the same arc in a circle. Find the unknown angle.

Answer: 65

Question 7: A real-world scenario: A circular garden has two benches placed such that they both subtend the same arc from the center. If one bench's angle at the center measures 80°, what can you say about the angle of the other bench?

Answer: The angle of the other bench at the center also measures 80° because they subtend the same arc.

Question 8: Construct two angles on the grid that are equal because they subtend the same arc, with one measuring 55°. Label your points clearly.

Answer: Students will draw and label; no numerical answer.

Question 9: Identify and correct the mistake in this reasoning: 'Two angles in a circle subtending different arcs can be equal.'

Answer: The statement is incorrect because angles subtending different arcs are not necessarily equal. Angles only equal if they subtend the same arc.

Question 10: Challenge: In a circle, an inscribed triangle has an angle of 75° at one vertex. If the opposite arc measures 150°, what are the measures of the other two angles?

Answer: 75° and 75°

Question 11: Using the Same Arc theorem, prove that two angles subtending the same arc are equal by reasoning or diagram.

Answer: Angles subtending the same arc are equal because they intercept the same segment of the circle, as shown in the diagram or explained logically.