Proofs with inscribed angles

Japanese school year: Junior high 3

What you learn

Learn how to prove geometric relationships using the inscribed angle theorem, its converse, and cyclic quadrilaterals. This builds on congruence and similarity proofs, combining angle relationships with circle geometry. These proofs are frequently used to solve intricate geometric problems involving circles and concyclic points.

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Key points

This theorem states that opposite angles $\angle A$ and $\angle C$ of a cyclic quadrilateral sum to $180^\circ$ . It helps find unknown angles and implies each exterior angle equals its opposite interior angle.

\angle A + \angle C = 180^\circ

Known as the tangent-chord theorem, this states that the angle $\angle BAT$ between tangent $AT$ and chord $AB$ equals the inscribed angle $\angle BCA$ subtended by that chord.

\angle BAT = \angle BCA

This theorem proves that four points are concyclic. If points $P$ and $Q$ lie on the same side of segment $AB$ and $\angle APB = \angle AQB$ , then points $A, B, P, Q$ lie on the same circle.

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Practice 1Coming soon