Angles in the Same Segment

Definition: A segment of a circle is the region between a chord and the arc it cuts off.

• The major segment is the region bounded by the chord and the major arc.
• The minor segment is the region bounded by the chord and the minor arc.

Theorem (Angles in the Same Segment):

Angles subtended by the same chord at points in the same segment are equal.

If points C and D are both on the major arc of chord AB, then:

∠ACB = ∠ADB

Converse:

If ∠ACB = ∠ADB (C, D on the same side of AB), then A, B, C, D are concyclic.

Important:

• Points must be in the same segment (same arc). Angles in different segments are NOT equal.
• The angle in the major segment is acute (when the chord is not a diameter).
• The angle in the minor segment is obtuse.
• The sum of angles in the major and minor segments (subtended by the same chord) is 180°.

Proof: Angles in the Same Segment are Equal

Given: Circle with centre O. Chord AB. Points C and D on the major arc.

To prove: ∠ACB = ∠ADB

Proof:

1. ∠AOB = 2 ∠ACB   (Central Angle Theorem: angle at centre = 2 × angle at C on the major arc) …(i)
2. ∠AOB = 2 ∠ADB   (Central Angle Theorem: angle at centre = 2 × angle at D on the major arc) …(ii)
3. From (i) and (ii): 2 ∠ACB = 2 ∠ADB
4. Therefore: ∠ACB = ∠ADB ■

Proof: Angles in Major and Minor Segments are Supplementary

Given: Point C on major arc, point E on minor arc of chord AB.

To prove: ∠ACB + ∠AEB = 180°

Proof:

1. ∠ACB = ∠AOB / 2   (C is on major arc) …(i)
2. Reflex ∠AOB = 360° − ∠AOB
3. ∠AEB = Reflex ∠AOB / 2 = (360 − ∠AOB) / 2 …(ii)
4. ∠ACB + ∠AEB = ∠AOB/2 + (360 − ∠AOB)/2 = 360/2 = 180° ■

Proof of the Converse (Concyclicity Test):

Given: ∠ACB = ∠ADB; C, D on the same side of line AB.

To prove: A, B, C, D lie on a circle.

Proof (by contradiction):

1. Assume D does NOT lie on the circle through A, B, C.
2. Let the circle through A, B, C intersect AD (or AD extended) at D′.
3. Then ∠AD′B = ∠ACB (angles in same segment of circle through A, B, C, D′).
4. But ∠ADB = ∠ACB (Given).
5. So ∠AD′B = ∠ADB, which is impossible unless D′ = D (exterior angle cannot equal interior angle).
6. Contradiction. Therefore D lies on the circle. ■

Key Configurations and Applications:

1. Multiple points on the same arc

• Any number of points on the same arc will all subtend the same angle with the same chord.
• If C, D, E, F are all on the major arc of AB: ∠ACB = ∠ADB = ∠AEB = ∠AFB.

2. Angles in the major segment

• All angles subtended at the major arc by the chord are equal and acute (if the chord is not a diameter).
• These angles are each half the central angle.

3. Angles in the minor segment

• All angles subtended at the minor arc are equal and obtuse.
• Each is half the reflex central angle.

4. Supplementary relationship

• An angle in the major segment + an angle in the minor segment = 180°.
• This is equivalent to the property of opposite angles of a cyclic quadrilateral.

5. Concyclicity test (converse)

• To prove four points are concyclic, show that two of them subtend equal angles at the other two (on the same side).
• This is a standard technique in geometry proofs.

Applications of Angles in the Same Segment:

• Proving concyclicity: The converse is the primary method for proving that four points lie on a circle. This is essential in advanced geometry proofs.
• Cyclic quadrilateral proofs: The property that opposite angles of a cyclic quadrilateral sum to 180° is a direct consequence of this theorem.
• Navigation: Two observers at different points on a circular coastline see the same pair of landmarks at the same angle, confirming the circular geometry.
• Structural engineering: Circular arches distribute loads. Points on the same arc experience equal angular relationships to the endpoints of a chord (support points).
• Circle construction: To draw a circle through three non-collinear points, the angles in the same segment property guarantees a unique circle exists.
• Optics: Points on a circular mirror at equal distances from the centre reflect light at equal angles, consistent with this theorem.