Circles • Topic 4 of 4

Angle Properties

The master rule for angles in a circle is that the angle subtended by an arc at the centre is exactly twice the angle it subtends at any point on the remaining circumference. From this single fact three workhorse results follow. Angles in the same segment are equal, because they subtend the same arc, so the centre angle (and hence the inscribed angle) is fixed. The angle in a semicircle is 90°, since the diameter gives a central angle of 180°, halving to 90° at the circumference — this is the fastest way to spot a right angle in CAT figures. The alternate segment theorem connects tangents to inscribed angles: the angle between a tangent and a chord drawn from the point of contact equals the inscribed angle in the alternate segment. In practice, when you see a tangent and a chord meeting, immediately mark the equal angle across the circle; when you see a diameter, mark the 90°. These reflexes turn multi-step problems into one or two lines.

✅ Solved examples

1. An arc subtends 130° at the centre of a circle. What angle does it subtend at a point on the major arc?
Angle at circumference = half the central angle = 130°/2 = 65°.
2. In a circle, AB is a diameter and C is a point on the circle. Find ∠ACB.
Angle in a semicircle is 90°, so ∠ACB = 90°.
3. A tangent at point A and a chord AB make an angle of 50°. Find the inscribed angle in the alternate segment subtended by AB.
By the alternate segment theorem, the inscribed angle equals the tangent–chord angle = 50°.
4. Two points C and D lie on the same side of chord AB and each subtends an angle on AB. If ∠ACB = 40°, find ∠ADB.
Angles in the same segment are equal, so ∠ADB = 40°.

✏️ Practice — try these, take hints as needed

1. An arc subtends 88° at the centre. Find the inscribed angle on the major arc.
Inscribed = half central.
88/2.
Same arc.
44°
2. PQ is a diameter; R is on the circle and ∠RPQ = 35°. Find ∠PQR.
∠PRQ = 90° (semicircle).
Angles of triangle sum to 180°.
180 − 90 − 35.
55°
3. A tangent–chord angle is 62°. Find the inscribed angle in the alternate segment.
Alternate segment theorem.
They are equal.
No subtraction needed.
62°
4. Inscribed angle subtending an arc is 47°. What is the central angle on the same arc?
Central = 2 × inscribed.
2 × 47.
Same arc.
94°
5. In the same segment, ∠ACB = (2x + 10)° and ∠ADB = (3x − 20)°. Find x.
Same segment ⇒ equal.
2x + 10 = 3x − 20.
Solve for x.
x = 30 (each angle = 70°)

📝 Topic test — 8 questions

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