Converse of Theorem 9.8

Theorem: If two angles subtended by the same segment (or chord) of a circle are equal, then the points defining these angles lie on the circumference of the same circle.

Given: A segment (or chord) AB has two points C and D on the same side of AB with ∠ACB = ∠ADB.

To Prove: Points A, B, C, and D are concyclic (i.e., they lie on the circumference of the same circle).

Proof: Consider the circle that passes through points A, B, and C. Let this circle be denoted as ω.

From the Inscribed Angle Theorem , ∠ACB is subtended by the chord AB and is an inscribed angle in the circle C1.

Since ∠ACB = ∠ADB and ∠ACB is an inscribed angle subtended by chord AB in circle C1, we can say that: ∠ADB must also subtend the same chord AB in circle C1.

This means that D also lies on the circle C1.

Therefore, points A, B, C, and D all lie on the circumference of the same circle, making them concyclic.