Question 1: In a cyclic quadrilateral, the opposite angles are 115° and x°. Calculate the value of x.

Answer: 65

Question 2: Two angles in a cyclic quadrilateral are 80° and 105°. Find the measure of the opposite angle to the 80° angle.

Answer: 75

Question 3: A quadrilateral inscribed in a circle has three angles measuring 70°, 85°, and 130°. What is the measure of the fourth angle?

Answer: 75

Question 4: Construct a cyclic quadrilateral where one pair of opposite angles are 60° and 120°. Show that these angles are supplementary.

Answer: Angles are supplementary because 60° + 120° = 180°

Question 5: In a circle, a quadrilateral is inscribed such that one angle is 45°. What is the measure of the opposite angle?

Answer: 135

Question 6: Two opposite angles in a cyclic quadrilateral are 90° and x°. Explain why x° must be 90°.

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

Question 7: A circle has a cyclic quadrilateral with angles 50° and 130°. Calculate the two missing angles, which are opposite to these angles.

Answer: 130 and 50

Question 8: A cyclic quadrilateral has an angle of 110°. What is the measure of its opposite angle? Show your reasoning.

Answer: 70

Question 9: Challenge: Prove that in any cyclic quadrilateral, the sum of each pair of opposite angles is 180°.

Answer: In a cyclic quadrilateral, angles subtended by the same arc are equal and supplementary, hence the sum of opposite angles is 180°.

Question 10: Error Analysis: A student claims that in a cyclic quadrilateral, all angles are equal. Correct the mistake and explain why.

Answer: The student is mistaken because opposite angles are supplementary, not necessarily equal. Equal angles occur only in special cases like squares.