Identification and Verification of Properties of Circle

📐 Experiment: Angles standing on the same arc
Figure∠ACB∠ADBResult
1.———
2.———
3.———
Conclusion:—
Theorem 2: The angles on the circumference of a circle, based on the same arc are equal.

Given:

O is the centre of a circle in which inscribed angles ∠ACB and ∠ADB are based on the same arc AB.

To prove:

∠ACB = ∠ADB

Construction: (Click to reveal)

Join centre O of the circle with the points A and B successively.

Proof: (Click to reveal table)

S. No.StatementsS. No.Reasons
1.∠AOB = 2∠ACB 1.Central angle = 2 × inscribed angle (same arc AB)
2.∠AOB = 2∠ADB 2.Central angle = 2 × inscribed angle (same arc AB)
3.2∠ACB = 2∠ADB
or, ∠ACB = ∠ADB		 3.From (1) and (2)
Proved.
∴ The angles on the circumference of a circle, based on the same arc are equal.