What is the property of angles subtended by the same arc?

Angles subtended by the same arc are equal.

When two or more angles are subtended by the same arc in a circle, they are always equal to each other. This is a fundamental property in circle geometry and is often used to solve various problems involving circles. To understand why this is the case, consider a circle with an arc AB. If you draw two angles, ∠ACB and ∠ADB, where points C and D lie on the circumference of the circle, both angles will be equal.

This property arises because the angles subtended by the same arc are essentially 'looking' at the same portion of the circle. The arc AB creates a specific 'slice' of the circle, and any angle formed by drawing lines from the endpoints of this arc to another point on the circumference will measure the same. This is due to the fact that the angles are inscribed angles, and the measure of an inscribed angle is always half the measure of the central angle that subtends the same arc.

For example, if the central angle ∠AOB (where O is the centre of the circle) measures 80 degrees, then any angle subtended by the arc AB, such as ∠ACB or ∠ADB, will measure 40 degrees. This property is very useful in solving problems related to circle theorems, as it allows you to identify and equate angles easily, simplifying the process of finding unknown angles or proving certain geometric properties.