Identification and Verification of Properties of Circle

📐 Experiment: Opposite angles of a cyclic quadrilateral sum to 180°
Figure∠DAB∠ABC∠BCD∠ADC∠DAB+∠BCD∠ABC+∠ADCResult
1.———————
2.———————
3.———————
Conclusion:—
Theorem: The opposite angles of a cyclic quadrilateral are supplementary (sum to 180°).

Given:

O is the centre of the circle. ABCD is a cyclic quadrilateral where all vertices lie on the circumference.

To prove:

∠ABC + ∠ADC = 180°
∠BCD + ∠BAD = 180°

Construction: (Click to reveal)

Join points A and C to the center O (Draw OA and OC).

Proof: (Click to reveal table)

S.N.StatementsS.N.Reasons
1.Obtuse ∠AOC = 2∠ADC 1.Central angle is double the inscribed angle on the same arc ABC.
2.Reflex ∠AOC = 2∠ABC 2.Central angle is double the inscribed angle on the same arc ADC.
3.Obtuse ∠AOC + Reflex ∠AOC = 2∠ADC + 2∠ABC 3.Adding statement (1) and (2).
4.360° = 2(∠ADC + ∠ABC) 4.The sum of angles around a point (O) is 360°.
5.∴ ∠ABC + ∠ADC = 180° 5.Dividing both sides by 2.
Proved.
Similarly, we can prove ∠BCD + ∠BAD = 180°.