Question 1: In a cyclic quadrilateral, two opposite angles are 65° and 115°. Are these angles correctly identified as opposite angles? Explain your answer.

Answer: Yes, because opposite angles in a cyclic quadrilateral are supplementary, and 65° + 115° = 180°.

Question 2: Calculate the measure of an angle opposite a 70° angle in a cyclic quadrilateral.

Answer: 110

Question 3: True or False: Opposite angles in a cyclic quadrilateral are always equal.

Answer: False

Question 4: If one pair of opposite angles in a quadrilateral are both 90°, is the quadrilateral necessarily cyclic? Explain your reasoning.

Answer: No, because right angles alone do not guarantee the quadrilateral is cyclic; it needs to satisfy the supplementary angle condition.

Question 1: In a cyclic quadrilateral, the measure of one angle is 120°. What are the measures of its opposite angles? Justify your answer.

Answer: 60° each, because opposite angles in a cyclic quadrilateral are supplementary (120° + 60° = 180°).

Question 2: A quadrilateral has angles measuring 80°, 100°, 60°, and x°. If the quadrilateral is cyclic, find x and explain your reasoning.

Answer: x = 120°, because the sum of internal angles is 360°, so 80 + 100 + 60 + x = 360, thus x = 120°, and opposite angles are supplementary.

Question 3: Construct a cyclic quadrilateral where two opposite angles are 85° and 95°. Calculate the other two angles and verify their correctness.

Answer: The other two angles are 85° and 95°, opposite angles are supplementary, so each pair sums to 180°. Since 85° and 95° are opposite, they satisfy the condition.

Question 1: A circular table has a quadrilateral pattern painted on it. If two opposite angles where the pattern meets are 70° and 110°, are these angles correctly positioned as opposite angles? Explain why or why not.

Answer: Yes, because opposite angles in a cyclic quadrilateral are supplementary, and 70° + 110° = 180°.

Question 1: Prove that in any cyclic quadrilateral, the opposite angles are supplementary using circle theorems.

Answer: In a cyclic quadrilateral, all vertices lie on a circle. The measure of an angle subtended by a chord is half the measure of the intercepted arc. Opposite angles intercept supplementary arcs, so their measures sum to 180°, making them supplementary.

Question 2: A cyclic quadrilateral has one angle measuring 100°. What is the measure of the angle opposite to it? Explain your reasoning.

Answer: 80°, because opposite angles are supplementary in a cyclic quadrilateral (100° + x° = 180°).

Question 1: Identify and correct the mistake in the following statement: 'In any quadrilateral, if two angles are 90°, then the angles are opposite.'

Answer: The mistake is assuming the two 90° angles are opposite; in general, they could be adjacent unless the quadrilateral is cyclic and the angles are opposite in a rectangle.