Find angles by combining the inscribed angle theorem with other properties, such as cyclic quadrilaterals and tangents.
Property of a Cyclic Quadrilateral
In a quadrilateral inscribed in a circle (a cyclic quadrilateral), the sum of a pair of opposite angles is 180°.
📘 Example ①
In cyclic quadrilateral ABCD, find ∠C when ∠A=75°.
Answer: Since ∠A + ∠C = 180°, ∠C = 180° - 75° = 105°
In cyclic quadrilateral ABCD, find ∠C when ∠A=75°.
Answer: Since ∠A + ∠C = 180°, ∠C = 180° - 75° = 105°
📘 Example ②
An inscribed angle subtending arc AB is 40°. Find ∠ACB, which is another inscribed angle subtending the same arc.
Answer: All inscribed angles subtending the same arc are equal, so ∠ACB = 40°
An inscribed angle subtending arc AB is 40°. Find ∠ACB, which is another inscribed angle subtending the same arc.
Answer: All inscribed angles subtending the same arc are equal, so ∠ACB = 40°
💡 Key Point
- Cyclic quadrilateral: sum of opposite angles = 180°
- All inscribed angles subtending the same arc are equal
Practice Problems
- In cyclic quadrilateral ABCD, find ∠D when ∠B=110°.
Answers and Explanations
- Answer: 70°. ∠B + ∠D = 180° → ∠D = 70°.