Question 1: In a cyclic quadrilateral, the measure of one angle is 110°. What is the measure of the opposite angle?

Answer: 70

Question 2: Two opposite angles in a cyclic quadrilateral are 85° and 95°. Are these angles correctly positioned? Explain your reasoning.

Answer: Yes, since opposite angles in a cyclic quadrilateral are supplementary, their sum is 180°, and 85° + 95° = 180°.

Question 3: Calculate the missing angle in a cyclic quadrilateral if three angles are 60°, 80°, and 110°.

Answer: 110

Question 4: A quadrilateral inscribed in a circle has two opposite angles measuring 120° and x°. Find x.

Answer: 60

Question 5: Construct a quadrilateral with one angle of 75° and its opposite angle of 105°. Show that these angles are supplementary.

Answer: 75° + 105° = 180°, confirming they are supplementary.

Question 6: In a cyclic quadrilateral, if one angle measures 130°, what are the possible measures of its opposite angle?

Answer: 50

Question 7: A cyclic quadrilateral has angles of 100°, 80°, and 70°. Calculate the fourth angle and verify if the opposite angles are supplementary.

Answer: 110°, and 70° + 110° = 180° (yes, supplementary)

Question 8: A farmer designs a circular garden with four sections, each section's corner angles are inscribed in the circle. If one section's angle is 90°, what is its opposite angle? Justify your answer.

Answer: 90°, because opposite angles in a cyclic quadrilateral are supplementary and 90° + 90° = 180°.

Question 9: Identify and correct the mistake in this statement: 'In a cyclic quadrilateral, the opposite angles are always equal.'

Answer: The mistake is that opposite angles are supplementary, not necessarily equal.

Question 10: Challenge: Prove that in a cyclic quadrilateral, if one pair of opposite angles are both 70°, then the other pair must be 110° each.

Answer: Since opposite angles are supplementary, 180° - 70° = 110°, so the other pair must also be 110° each.