Inscribed Angles (Angles in the Same Segment)

1. What is an Inscribed Angle?

An inscribed angle is an angle formed on the circumference of a circle by two chords.

If points \(A, B, C\) lie on a circle and the angle is made at \(B\), then \(\angle ABC\) is an inscribed angle.

The key feature is that the vertex lies on the circle, not at the centre.

Every inscribed angle subtends (intercepts) an arc on the opposite side of the circle.

For angle \(\angle ABC\), the intercepted arc is \(AC\).

The arc and the angle always correspond to each other.

This is the most important fact about inscribed angles:

An inscribed angle is half the measure of the central angle that subtends the same arc.

\(\angle ABC = \dfrac{1}{2} \angle AOC\)

where \(O\) is the centre and \(AC\) is the intercepted arc.

Another very important property is that:

All inscribed angles that intercept the same arc are equal.

So if angles \(\angle ABC\) and \(\angle ADC\) subtend arc \(AC\), then:

\(\angle ABC = \angle ADC\)

This property is used in many geometry proofs, especially when identifying equal angles in circle diagrams.

If an inscribed angle subtends a diameter, then the angle is always a right angle.

This is called the Thales' Theorem.

\(\angle ABC = 90^\circ\)

where \(AC\) is the diameter.

Here are simple numerical examples that help you visualise inscribed angle rules.

In a circle, the central angle subtending arc \(AC\) is \(80^\circ\). Then the inscribed angle subtending the same arc is:

\angle ABC = \dfrac{1}{2} \times 80^\circ = 40^\circ

If \(\angle ABC = 45^\circ\) and both angles intercept arc \(AC\), then:

\angle ADC = 45^\circ

Inscribed angles appear in structures such as:

They help in designing symmetric curved shapes.