Why This Topic Is Worth So Many Marks#
Cyclic quadrilaterals, tangents, and the tan-chord theorem combine with the core theorems from the previous page to form the full toolkit for circle geometry proofs. In Paper 2, circle geometry questions range from 35–50 marks. Most of those marks come from multi-step proofs that chain several theorems together.
The key to success: know the exact reason phrase for each theorem and practise combining them.
1. Cyclic Quadrilaterals#
A cyclic quadrilateral is any four-sided shape where all four vertices lie on the circle. The circle is called the circumscribed circle.
Theorem: Opposite Angles Are Supplementary#
The opposite angles of a cyclic quadrilateral add to $180°$.
Reason to write: opp $\angle$s of cyclic quad
$$\hat{A} + \hat{C} = 180° \qquad \text{and} \qquad \hat{B} + \hat{D} = 180°$$Why It Works#
Each pair of opposite angles stands on arcs that together make up the full circle ($360°$). Since an inscribed angle equals half the arc it stands on, the two opposite angles together equal half of $360° = 180°$.
Theorem: Exterior Angle of a Cyclic Quad#
The exterior angle of a cyclic quadrilateral equals the interior opposite angle.
Reason to write: ext $\angle$ of cyclic quad
This follows directly from the opposite angles theorem: if $\hat{A} + \hat{C} = 180°$ and $\hat{A} + \hat{A}_{\text{ext}} = 180°$ (angles on a straight line), then $\hat{A}_{\text{ext}} = \hat{C}$.
Worked Example 1 — Finding Angles#
In cyclic quad ABCD: $\hat{A} = 75°$ and $\hat{B} = 110°$.
| Statement | Reason |
|---|---|
| $\hat{C} = 180° - 75° = 105°$ | opp $\angle$s of cyclic quad |
| $\hat{D} = 180° - 110° = 70°$ | opp $\angle$s of cyclic quad |
Check: $75 + 110 + 105 + 70 = 360°$ ✓
Worked Example 2 — Finding an Unknown#
ABCD is a cyclic quad. $\hat{A} = 2x + 10°$ and $\hat{C} = 3x - 20°$. Find $x$ and all angles.
| Statement | Reason |
|---|---|
| $\hat{A} + \hat{C} = 180°$ | opp $\angle$s of cyclic quad |
| $(2x + 10) + (3x - 20) = 180$ | |
| $5x - 10 = 180$ | |
| $x = 38°$ |
$\hat{A} = 2(38) + 10 = 86°$ and $\hat{C} = 3(38) - 20 = 94°$
Check: $86 + 94 = 180°$ ✓
2. Proving a Quadrilateral IS Cyclic (The Converses)#
This is one of the most commonly tested skills. You must know three ways to prove a quadrilateral is cyclic:
| Method | What to show | Reason to write |
|---|---|---|
| Converse 1 | Opposite angles are supplementary ($\hat{A} + \hat{C} = 180°$) | opp $\angle$s suppl / converse: opp $\angle$s of cyclic quad |
| Converse 2 | An exterior angle equals the interior opposite angle | ext $\angle$ = int opp $\angle$ / converse: ext $\angle$ of cyclic quad |
| Converse 3 | Two points on the same side of a line subtend equal angles to the line | $\angle$s in same segment / converse: $\angle$s in same segment |
Worked Example 3 — Proving Cyclic#
In quadrilateral PQRS, $\hat{P} = 80°$ and $\hat{R} = 100°$. Prove PQRS is a cyclic quadrilateral.
| Statement | Reason |
|---|---|
| $\hat{P} + \hat{R} = 80° + 100° = 180°$ | |
| $\therefore$ PQRS is a cyclic quad | opp $\angle$s supplementary |
Worked Example 4 — Proving Cyclic Using Equal Angles#
In the diagram, $\hat{A}_1 = \hat{D}_1$ and both angles are subtended by line BC on the same side. Prove ABCD is cyclic.
| Statement | Reason |
|---|---|
| $\hat{A}_1 = \hat{D}_1$ | Given |
| A and D are on the same side of BC | From diagram |
| $\therefore$ ABCD is a cyclic quad | converse: $\angle$s in same segment |
3. Tangent Theorems#
A tangent is a line that touches the circle at exactly one point (the point of tangency/contact).
Theorem A: Radius ⊥ Tangent#
A radius drawn to the point of tangency is perpendicular to the tangent.
Reason: rad $\perp$ tan
Theorem B: Equal Tangents from an External Point#
Two tangents drawn from the same external point are equal in length.
Reason: tans from same pt
Why Equal Tangents Works#
Draw the two radii to the tangent points and the line from the centre to the external point. This creates two right-angled triangles (rad $\perp$ tan). They share the hypotenuse (centre to external point) and have equal sides (both radii). By RHS congruence, the triangles are congruent, so the tangent lengths are equal.
Worked Example 5 — Tangent Length Problem#
Tangents PA and PB are drawn from external point P to a circle with centre O. $PA = 12$ cm and $OA = 5$ cm. Find OP.
| Statement | Reason |
|---|---|
| $O\hat{A}P = 90°$ | rad $\perp$ tan |
| $OP^2 = OA^2 + PA^2 = 25 + 144 = 169$ | Pythagoras |
| $OP = 13$ cm | |
| $PB = PA = 12$ cm | tans from same pt |
Worked Example 6 — Tangent with Angle#
TA is a tangent to the circle at A. O is the centre. $\hat{T} = 30°$. Find $O\hat{A}T$ and $A\hat{O}T$.
| Statement | Reason |
|---|---|
| $O\hat{A}T = 90°$ | rad $\perp$ tan |
| $A\hat{O}T = 180° - 90° - 30° = 60°$ | $\angle$ sum of $\triangle$ |
4. The Tan-Chord Theorem#
This is the theorem students find hardest to spot in diagrams, but it’s extremely powerful.
The angle between a tangent and a chord at the point of contact equals the inscribed angle subtended by the same chord in the alternate segment.
Reason: tan-chord theorem
Understanding “Alternate Segment”#
The chord divides the circle into two segments. The tangent touches the circle at one end of the chord. The “alternate segment” is the segment on the other side of the chord from the tangent-chord angle.
Worked Example 7 — Basic Tan-Chord#
TA is a tangent at A. Chord AB makes an angle of $40°$ with the tangent. C is a point on the major arc. Find $\hat{C}$.
| Statement | Reason |
|---|---|
| $\hat{C} = 40°$ | tan-chord theorem (angle in alternate segment) |
Worked Example 8 — Tan-Chord in a Proof#
TA is a tangent at A. $\hat{A}_1 = 50°$ (between tangent and chord AC). B is on the minor arc. Find $\hat{B}$.
The alternate segment for $\hat{A}_1$ is the major arc side. So $\hat{C}_{\text{major}} = 50°$.
But B is on the minor arc. The angle at B and the angle on the major arc are related:
| Statement | Reason |
|---|---|
| Let $\hat{D} = 50°$ (D on major arc) | tan-chord theorem |
| $\hat{B} + \hat{D} = 180°$ | opp $\angle$s of cyclic quad ABDC |
| $\hat{B} = 180° - 50° = 130°$ |
5. Multi-Step Proofs — Combining Theorems#
Exam proofs almost always require chaining 3–5 theorems together. Here’s the approach:
Strategy#
- Mark the diagram — fill in all given angles, equal sides (radii!), right angles (rad ⊥ tan)
- Identify isoceles triangles — every pair of radii creates one
- Look for the theorems — which pattern matches?
- Write Statement | Reason for every single line
- Work toward the goal — what do they want you to prove?
Worked Example 9 — Full Multi-Step Proof#
O is the centre. TA is a tangent at A. B and C are on the circle. $O\hat{A}B = 25°$. Prove that $\hat{C} = 65°$.
| Statement | Reason |
|---|---|
| $O\hat{A}T = 90°$ | rad $\perp$ tan |
| $T\hat{A}B = 90° - 25° = 65°$ | (since $O\hat{A}T = O\hat{A}B + B\hat{A}T$) |
| $\hat{C} = T\hat{A}B = 65°$ | tan-chord theorem |
Worked Example 10 — Proving Parallel Lines#
O is the centre. TA is a tangent at A. $\hat{A}_1 = x$ (tan-chord angle). $B\hat{O}A = 2x$ at the centre. Prove TA $\parallel$ OB.
| Statement | Reason |
|---|---|
| $\hat{A}_1 = x$ | Given |
| $\hat{C} = x$ | tan-chord theorem (angle in alternate segment) |
| $B\hat{O}A = 2x$ | Given |
| $B\hat{O}A = 2\hat{C}$ | $\angle$ at centre = $2 \times \angle$ at circumf |
| This is consistent ✓ | |
| $O\hat{A}B = O\hat{B}A$ | base $\angle$s of isos $\triangle$ ($OA = OB$ = radii) |
| In $\triangle OAB$: $2 \times O\hat{A}B + 2x = 180°$ | $\angle$ sum of $\triangle$ |
| $O\hat{A}B = 90° - x$ | |
| $T\hat{A}O = O\hat{A}B + \hat{A}_1 = (90° - x) + x = 90°$ | |
| But $O\hat{A}T = 90°$ | rad $\perp$ tan |
| $O\hat{B}A = 90° - x$ | base $\angle$ of isos $\triangle$ |
| $\hat{A}_1 = x$ and $O\hat{B}A = 90° - x$ | Need alt angles… |
Actually, let’s use a cleaner approach:
| Statement | Reason |
|---|---|
| $\hat{A}_1 = x$ | Given (angle between tangent TA and chord AB) |
| $O\hat{B}A = O\hat{A}B$ | base $\angle$s of isos $\triangle$, $OA = OB$ (radii) |
| $A\hat{O}B = 2x$ | Given |
| $O\hat{A}B = O\hat{B}A = \frac{180° - 2x}{2} = 90° - x$ | $\angle$ sum of $\triangle$ |
| $T\hat{A}B = \hat{A}_1 = x$ | Given |
| $T\hat{A}B + O\hat{A}B = x + (90° - x) = 90° = T\hat{A}O$ | |
| $T\hat{A}O = 90°$ | rad $\perp$ tan ✓ (confirms consistency) |
| Now: $\hat{A}_1 = x$ (alt angle with TA) and $O\hat{B}A = 90° - x$ |
For the parallel proof, we need co-interior or alternate angles. If $\hat{A}_1 = O\hat{B}A$, then TA $\parallel$ OB (alt $\angle$s equal). This requires $x = 90° - x$, i.e., $x = 45°$. So the general parallel proof requires additional given information.
Worked Example 11 — Extended Proof with Cyclic Quad#
ABCD is a cyclic quad. TA is a tangent at A. $T\hat{A}D = 55°$ and $A\hat{B}C = 110°$. Find $\hat{D}_1$ (the angle ADC).
| Statement | Reason |
|---|---|
| $\hat{D}_1 = 180° - \hat{B} = 180° - 110° = 70°$ | opp $\angle$s of cyclic quad |
| Alternative: $T\hat{A}D = 55°$ | Given |
| $A\hat{C}D = T\hat{A}D = 55°$ | tan-chord theorem |
6. The Complete Theorem Reference#
Memorise these exact phrases — they are what markers look for:
| Theorem | Reason to write |
|---|---|
| Centre angle = 2× circumference | $\angle$ at centre = $2 \times \angle$ at circumf |
| Diameter gives 90° | $\angle$ in semi-circle |
| Same segment gives equal angles | $\angle$s in same segment |
| Cyclic quad: opp angles = 180° | opp $\angle$s of cyclic quad |
| Cyclic quad: ext angle = int opp | ext $\angle$ of cyclic quad |
| Tan-chord angle | tan-chord theorem |
| Radius ⊥ tangent | rad $\perp$ tan |
| Equal tangents from ext point | tans from same pt |
| Perpendicular bisects chord | line from centre $\perp$ to chord |
| Isoceles triangle (radii) | radii / base $\angle$s of isos $\triangle$ |
Converses (for proving things)#
| To prove… | Show that… | Reason |
|---|---|---|
| Quad is cyclic | Opposite angles sum to 180° | converse: opp $\angle$s suppl |
| Quad is cyclic | Ext angle = int opp angle | converse: ext $\angle$ = int opp $\angle$ |
| Quad is cyclic | Equal angles on same side of line | converse: $\angle$s in same segment |
| Line is a tangent | Line ⊥ to radius at the point on circle | converse: rad $\perp$ tan |
| Line is a diameter | Angle at circumference = 90° | converse: $\angle$ in semi-circle |
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| Writing “angles in circle” as a reason | This is too vague — markers need the specific theorem | Use the exact phrases from the reason bank above |
| Confusing theorem and converse | “Opp angles supplementary” finds angles in a known cyclic quad. The converse proves the quad IS cyclic | Know which direction you’re working: theorem → find angles; converse → prove cyclic |
| Forgetting to state parallel lines | When using alternate or co-interior angles, you must explicitly state which lines are parallel | Write: “$AB \parallel CD$ (given)” before using alt angles |
| Not identifying the alternate segment | In tan-chord, the angle equals the inscribed angle in the segment on the other side of the chord | Draw the chord and identify which side the tangent angle is on; look on the opposite side |
| Assuming angles are equal from the diagram | Never assume — you must prove every claim with a theorem | Even if it “looks” like 90°, you must state the reason |
| Missing the isoceles triangle | Two radii in the same triangle → isoceles → base angles equal | Every time O connects to two points on the circle, look for the isoceles triangle |
| Not checking angle sums | Angles in a triangle = 180°; angles on a line = 180°; angles around a point = 360° | Use these as verification checks throughout your proof |
💡 Pro Tips for Exams#
1. The “Mark the Diagram” Ritual#
Before writing a single line of proof:
- ✅ Mark all radii with tick marks (→ isoceles triangles)
- ✅ Mark all right angles at tangent points (rad ⊥ tan)
- ✅ Fill in all given angles
- ✅ Mark equal angles (same segment, base angles, tan-chord)
- ✅ Identify cyclic quads (four points on the circle)
2. The “Reason Bank” Strategy#
Write every theorem reason on one page before the exam. During the exam, for each line of your proof, scan the bank and pick the matching reason. This prevents leaving out reasons (which costs marks).
3. Proof Structure Template#
Statement | Reason
---------------------------------- | ---------------------------
[What you know] | Given
[Intermediate step] | [Specific theorem]
[Another step] | [Specific theorem]
∴ [What you needed to prove] |4. The Three Most Common Proof Chains#
- Tan-chord → same segment → conclusion: Start with the tangent-chord angle, transfer it to an inscribed angle, then use it elsewhere.
- Isoceles triangle → angle at centre → angle at circumference: Use radii to find base angles, combine to find the centre angle, then halve for the circumference angle.
- Opposite angles → prove cyclic → use cyclic quad properties: Show opposite angles are supplementary to prove the quad is cyclic, then use cyclic quad theorems.
🔗 Related Grade 11 topics:
- Core Circle Theorems — you MUST know these before tackling cyclic quads and tangents
- Analytical Geometry: Circles & Tangents — the algebraic approach to circles and tangent lines