Geometry & Mensuration • Topic 3 of 5

Circles

Key circle theorems: the angle at the centre is twice the angle at the circumference on the same arc; the angle in a semicircle is 90 degrees; equal chords are equidistant from the centre; the tangent is perpendicular to the radius at the point of contact; and tangents drawn from an external point are equal. Cyclic-quadrilateral opposite angles sum to 180.

✅ Solved examples

1. Angle at centre is 80 degrees on an arc. Angle at circumference?
Half: 40 degrees.
2. Angle in a semicircle?
90 degrees.
3. A cyclic quadrilateral has one angle 95 degrees. Its opposite angle?
180 - 95 = 85 degrees.
4. A tangent meets a radius at the point of contact. The angle between them?
90 degrees.

✏️ Practice — try these, take hints as needed

1. Centre angle 110 -> circumference angle?
Half.
—
—
55 degrees
2. Cyclic quad angle 70 -> opposite?
Sum 180.
180 - 70.
—
110 degrees
3. Two tangents from an external point compare how?
Theorem.
—
—
Equal in length
4. Angle in a semicircle?
Fixed.
—
—
90 degrees
5. Circumference angle 30 -> centre angle?
Twice.
—
—
60 degrees

📝 Topic test — 8 questions

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