Solve problems by combining the inscribed angle theorem, cyclic quadrilaterals, and the conditions for points to be concyclic.
📘 Example ①
Points A, B, C, and D lie on a circle, and chords AC and BD intersect at P. What can be said when ∠ABD=35° and ∠ACD=35°?
Answer: The inscribed angles subtending arc AD are ∠ABD = ∠ACD = 35° (inscribed angles subtending the same arc). This holds naturally.
Points A, B, C, and D lie on a circle, and chords AC and BD intersect at P. What can be said when ∠ABD=35° and ∠ACD=35°?
Answer: The inscribed angles subtending arc AD are ∠ABD = ∠ACD = 35° (inscribed angles subtending the same arc). This holds naturally.
💡 Key Point
- Inscribed angle theorem: same arc → equal inscribed angles
- Converse of the theorem: equal inscribed angles → concyclic points
- Cyclic quadrilateral: sum of opposite angles = 180°
Practice Problems
- Find the inscribed angle subtending an arc whose central angle is 160°.
- In a cyclic quadrilateral, find the opposite angle when one angle is 85°.
Answers and Explanations
- Answer: 80°. 160 ÷ 2 = 80.
- Answer: 95°. 85° + 95° = 180°.