Equal Chords, Equal Arcs, Equal Angles

Statement

In a circle, there is a close relationship between chords, arcs, and the angles at the centre. The rule is:

\[ \text{Equal chords} \;\Longleftrightarrow\; \text{equal arcs} \;\Longleftrightarrow\; \text{equal angles at the centre} \]

This means if any two of these are equal, then the third is also equal.

Why it’s true

• A chord divides the circle into two arcs.
• If two chords are equal in length, they subtend equal arcs at the circumference.
• The arc length is directly linked to the angle at the centre: a larger arc gives a larger angle, a smaller arc a smaller angle.
• Therefore, equal chords ⇔ equal arcs ⇔ equal central angles.

Recipe (how to use it)

1. Identify the chord(s) in question.
2. Check if two chords are equal in length.
3. If yes, conclude their arcs are equal, and so are the angles they subtend at the centre.
4. Work backwards too: if central angles are equal, then the corresponding chords and arcs are also equal.

Spotting it

This rule applies in circle geometry problems when equal chords, equal arcs, or equal central angles are mentioned. It is often used to link one type of equality to another.

Common pairings

• Circle theorems (angle at centre, angle at circumference).
• Congruent isosceles triangles (formed by radii and chords).
• Proofs involving symmetry in circles.

Mini examples

1. Given: Two chords AB and CD are equal. Conclude: The arcs subtended by AB and CD are equal, and so are the central angles ∠AOB and ∠COD.
2. Given: Arc PQ = Arc RS. Conclude: Chords PQ and RS are equal, and ∠POQ = ∠ROS.

Pitfalls

• Confusing the angle at the centre with the angle at the circumference (centre angle is double the angle at circumference).
• Assuming unequal chords must always correspond to unequal arcs—be careful with semicircles.

Exam strategy

• When working with equal chords or arcs, always mark the corresponding central angles.
• Use symmetry in circle diagrams to check your reasoning.
• Link this rule with the “angle at the centre is twice the angle at the circumference” theorem for multi-step problems.

Summary

Equal chords in a circle subtend equal arcs, and these arcs subtend equal central angles. The three properties are fully interchangeable, making this rule a key link in circle geometry.