Geometry and measuresGCSE Higher / IGCSE Extended

In the same circle, equal chords subtend equal angles at the centre and equal minor arcs

Recognise the theorem, follow two examples, then try five questions.

  • The converse also holds. The result also applies to congruent circles, with corresponding minor arcs.

The theorem

In the same circle, equal chords subtend equal angles at the centre and equal minor arcs.

Example 1

With numerical values

is the centre and chords and are equal. Find the angle .

Equal chords subtend equal angles at the centre.

Example 2

With algebra

is the centre and chords and are equal. Find . Each labelled expression is an angle in degrees.

Equal chords subtend equal angles at the centre.Collect like terms.
Substitute the value of .
Labelled angles

Your turn

Question 1

The marked chords are equal. Find angle .

Check answer
Equal chords subtend equal angles at the centre.
Question 2

The marked chords are equal. Find the value of .

Check answer
Equal chords subtend equal angles at the centre.Collect like terms.
Substitute the value of .
Labelled angles
Question 3

The marked chords are equal. Find angle .

Check answer
Equal chords subtend equal angles at the centre.
Question 4

The marked chords are equal. Find the value of .

Check answer
Equal chords subtend equal angles at the centre.Collect like terms.
Substitute the value of .
Labelled angles
Question 5

The marked chords are equal. Find angle .

Check answer
Equal chords subtend equal angles at the centre.
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