Circle Theorems

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Notes

Angles at Centre & Circumference

The **angle at the centre** is **twice** the **angle at the circumference** subtended by the same arc.
Both angles are formed from the same two points on the circumference.
The theorem applies even when the triangle parts overlap or form a diamond shape.
Use the **reflex angle** at the centre when the angle at the circumference is obtuse.
Common mistake: confusing this theorem with opposite angles in a cyclic quadrilateral.

Angle in a Semicircle

The **angle in a semicircle** is **90°**.
This is a special case of the centre-circumference theorem (angle on diameter $= 180^{\circ}, halved)$ .
The right angle is opposite the **diameter**.
All three vertices must lie on the circumference, with one side as the diameter.
Often used with Pythagoras' theorem to find lengths.

Theorems with Chords

A **radius** that is **perpendicular** to a chord **bisects** the chord.
Conversely, the perpendicular bisector of a chord passes through the centre.
Equal chords are **equidistant** from the centre.
Chords and radii often form **isosceles triangles**.

Theorems with Tangents

A **radius** and a **tangent** meet at **right angles** (90°).
Tangents from an **external point** are **equal in length**.
Two tangents from the same point form a **kite** with two right angles.
Use Pythagoras or trigonometry in the right triangles formed.

Angles in Cyclic Quadrilaterals

A **cyclic quadrilateral** has all four vertices on the circumference.
**Opposite angles** in a cyclic quadrilateral **add up to 180°**.
This theorem does **not** apply to quadrilaterals with vertices not all on the circle.
Mark all angles on the diagram to help find relationships.

Angles in the Same Segment

**Angles in the same segment** are **equal**.
They are subtended by the same chord.
Look for a 'bowtie' shape with two angles on the same side of the chord.
The theorem works both ways: angles at either end of the chord are also equal.

The Alternate Segment Theorem

The angle between a **chord** and a **tangent** equals the angle in the **alternate segment**.
The alternate segment is the region on the opposite side of the chord from the angle.
Identify a cyclic triangle with one vertex touching the tangent.
The equal angle is inside the triangle opposite the side that forms the first angle.

General Tips

Always give a **reason** for each angle found (quote the circle theorem or angle fact).
Look for **isosceles triangles** formed by radii and chords.
Add **radii** and **right angles** to the diagram to reveal relationships.
Use properties of triangles, quadrilaterals, and parallel lines alongside circle theorems.

Angle at Centre is Twice Angle at Circumference

Angle in a Semicircle is 90°

Radius Perpendicular to Chord Bisects It

Alternate Segment Theorem

Practice questions

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1.In a circle, the angle at the centre is twice the angle at the circumference. If the angle at the circumference is 35°, what is the angle at the centre?
Easy
- A70°
- B35°
- C17.5°
- D140°
2.What is the angle in a semicircle?
Easy
- A90°
- B180°
- C45°
- D60°
3.A radius and a tangent meet at what angle?
Easy
- A90°
- B180°
- C45°
- D0°
4.Opposite angles in a cyclic quadrilateral add up to:
Easy
- A180°
- B90°
- C360°
- D270°
5.In the diagram, A, B, C, D lie on a circle with centre O. EA is a tangent at A. Angle $EAB = 61^{\circ}$ , angle $BAC = 55^{\circ}$ . Find angle BAO.
Medium
- A29°
- B61°
- C55°
- D35°
6.A, B, C, D are points on a circle, centre O. DOB is a straight line. Angle $DAC = 58^{\circ}$ . Find angle CDB.
Medium
- A58°
- B32°
- C29°
- D122°
7.P, Q, R are points on a circle, centre O. PO is parallel to QR and angle $POQ = 48^{\circ}$ . Find angle OPR.
Medium
- A24°
- B48°
- C12°
- D36°
8.A, B, C, D, E lie on a circle, centre O. Angle $AEB = 35^{\circ}$ , angle $ODE = 28^{\circ}$ , angle $ACD = 109^{\circ}$ . Find angle EBD.
Hard
- A35°
- B28°
- C109°
- D17°

Summary

## Circle Theorems – Quick Revision ### 1. Angle at Centre & Circumference - **Theorem:** The angle at the centre is twice the angle at the circumference (subtended by the same arc). - Look for two radii and lines to a point on circumference (arrowhead shape). - Remember to use the reflex angle at centre when needed. ### 2. Angle in a Semicircle - **Theorem:** The angle in a semicircle is 90°. - If a triangle has a diameter as one side, the angle opposite the diameter is 90°. - Special case of the centre & circumference theorem (180° ÷ 2 = 90°). ### 3. Theorems with Chords & Tangents - **Chord theorem:** The perpendicular bisector of a chord passes through the centre. (A radius bisects a chord at right angles.) - **Tangent-radius theorem:** A radius and a tangent meet at right angles (90°). - **Tangents from an external point:** Two tangents from the same external point are equal in length. ### 4. Cyclic Quadrilaterals - **Theorem:** Opposite angles in a cyclic quadrilateral add up to 180°. - A cyclic quadrilateral has all four vertices on the circumference. ### 5. Angles in the Same Segment - **Theorem:** Angles in the same segment are equal. - Look for a bowtie shape: two angles formed from the same chord to different points on the circumference are equal. ### 6. Alternate Segment Theorem - **Theorem:** The angle between a chord and a tangent is equal to the angle in the alternate segment. - The alternate segment is the region opposite the chord from the tangent point. - Often involves a cyclic triangle with one vertex touching a tangent. ### Exam Tips - Always give a reason for each angle you find (quote the theorem). - Mark equal angles and right angles on the diagram as you work. - Look for isosceles triangles formed by radii and chords. - Use other angle facts (angles in a triangle sum to 180°, angles on a straight line, etc.).

Key terms

Angle at centre is twice angle at circumference: The angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference.
Angle in a semicircle: The angle in a semicircle is 90°.
Perpendicular bisector of a chord: The perpendicular bisector of a chord passes through the centre of the circle.
Radius and tangent: A radius and a tangent meet at right angles (90°).
Tangents from an external point: Tangents from an external point are equal in length.
Cyclic quadrilateral: A quadrilateral with all four vertices on the circumference of a circle. Opposite angles sum to 180°.
Angles in the same segment: Angles in the same segment of a circle are equal.
Alternate segment theorem: The angle between a chord and a tangent is equal to the angle in the alternate segment.

Slides

Angle at Centre & Circumference

The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.
Look for two radii and lines from their endpoints to another point on the circumference.

If the angle at the centre is 80°, what is the angle at the circumference subtended by the same arc?

40°
80°
160°
20°

Angle in a Semicircle

The angle in a semicircle is always 90°.
This is a special case of the centre-circumference theorem: the diameter subtends 180° at the centre, so the angle at the circumference is 90°.

In a circle, a triangle has vertices on the circumference and one side is a diameter. What is the angle opposite the diameter?

45°
90°
60°
180°

Theorems with Chords & Tangents

A radius and a tangent meet at right angles (90°).
Tangents from an external point are equal in length.
The perpendicular bisector of a chord passes through the centre.

Which of these are true?

A radius and a tangent meet at 90°.
Two tangents from the same external point are equal in length.
A chord is always longer than the diameter.

Angles in Cyclic Quadrilaterals

A cyclic quadrilateral has all four vertices on the circumference.
Opposite angles in a cyclic quadrilateral add up to 180°.

In a cyclic quadrilateral, one angle is 110°. What is its opposite angle?

70°
110°
90°
55°

Angles in the Same Segment

Angles in the same segment of a circle are equal.
These angles are subtended by the same chord.

Angles in the same $____ are$ equal.

The Alternate Segment Theorem

The angle between a chord and a tangent is equal to the angle in the alternate segment.
The alternate segment is the region opposite the angle formed by the chord and tangent.

A tangent touches a circle at point A. Chord AB makes an angle of 50° with the tangent. What is the angle in the alternate segment?

50°
40°
130°
90°

Putting It All Together

When solving circle theorem problems, always state the theorem you are using.
Look for radii, diameters, tangents, chords, and cyclic quadrilaterals.
Remember that angles in a triangle sum to 180° and angles on a straight line sum to 180°.

Which statements are correct?

The angle at the centre is twice the angle at the circumference.
Opposite angles in a cyclic quadrilateral add up to 90°.
The angle in a semicircle is 90°.

Example: Worked Problem

In the diagram, A, B, C, D lie on a circle, centre O. EA is a tangent at A. Angle $EAB = 61^{\circ}$ , angle $BAC = 55^{\circ}$ .
Find angle BAO: Since OA is a radius and EA is a tangent, angle $OAE = 90^{\circ}$ . Angle $OAB = 90^{\circ} - 61^{\circ} = 29^{\circ}$ . Triangle OAB is isosceles $(OA=OB)$ , so angle $BAO = 29^{\circ}$ .

In the same diagram, angle $BAC = 55^{\circ}$ and angle $BAO = 29^{\circ}$ . What is angle OAC?

26°
55°
29°
84°

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