O Level MathematicsE4.8 Circle theorems (all properties of angles, chords, tangents, and cyclic quadrilaterals).

⭕🔢 Master the Circle! All Theorems Demystified

Edudent Academy
10 Jan 26

Circles pop up everywhere in O-Level exams, and the quickest route to full marks is a solid grip on circle theorems. From finding missing angles to proving lines are equal, these results turn messy diagrams into **straight-forward** puzzles. Let’s dive in!

Essential Theorems at a Glance

  • Angle subtended by an arc at the centre is twice the angle subtended at the circumference.
  • Angles in the same segment are equal.
  • Angle in a semicircle is 90∘90^\circ.
  • Opposite angles of a cyclic quadrilateral sum to 180∘180^\circ.
  • Tangent meets radius at 90∘90^\circ; the angle between a tangent and chord equals the angle in the opposite arc.

Worked Example: Tangent–Chord Angle

Problem: In circle OO, ABAB is a tangent at AA, and ACAC is a chord such that ∠CAB=38∘\angle CAB = 38^\circ. Point DD lies on the circle so that ADAD is a diameter. Find (i) ∠ACD\angle ACD, (ii) ∠ADC\angle ADC.

  • Step 1: By the **tangent–chord theorem**,
    ∠CAB=∠ACD=38∘.\angle CAB = \angle ACD = 38^\circ.
  • Step 2: Since ADAD is a diameter,
    ∠ADC=90∘\angle ADC = 90^\circ
    (angle in a semicircle).

Key Takeaways

Circle theorems act like shortcuts—**spot** the theorem, write the statement, and score the marks. Practise sketching quick, labelled diagrams and always mark right angles and equal arcs. Keep drilling past-paper questions and these facts will stick!