329. The Exterior Angle Secret · cyclic quads hide a beautiful equality

Try to break it

Drag A, B, C, or D. The exterior angle at B (∠CBE) always reads the same as the interior opposite angle ∠ADC. Why? Exterior + interior at B = 180° (linear pair), and ∠B + ∠D = 180° (cyclic) — so the exterior at B equals ∠D. Try to break the equality; impossible.

How you build it

Construct a cyclic quadrilateral with extended side.

• Mark a point O — the centre of the circle.
• Draw a circle centred on O — click O, then a point on the rim.
• Mark point A on the circle. Place A, B, C, D in order going around the circle.
• Mark point B on the circle, next to A going around.
• Mark point C on the circle, next around from B.
• Mark point D on the circle, completing the four points in order — a cyclic quadrilateral ABCD.
• Join A to B.
• Join B to C.
• Join C to D.
• Join D to A to close the cyclic quadrilateral.
• Draw a ray from A through B and beyond to E. The exterior angle ∠CBE equals the interior opposite angle ∠ADC.

The proof, step by step

Prove that the exterior angle of a cyclic quadrilateral equals the interior opposite angle.

• ABCD is a cyclic quadrilateral. (Given)
• ∠ABC + ∠ADC = 180° (Opposite angles of a cyclic quad are supplementary)
• ∠ABC + ∠CBE = 180° (Linear pair on straight line AE)
• ∴ ∠CBE = ∠ADC (Both are supplements of ∠ABC)

Worked example

In the given figure, ABCD is a cyclic quadrilateral. Side AB is produced to E. If ∠ADC = 110°, then ∠CBE is equal to:

By the exterior angle theorem for cyclic quadrilaterals, the exterior angle (∠CBE) is equal to the interior opposite angle (∠ADC). Hence, ∠CBE = 110°.

• 55°
• 70°
• 110° — correct
• 120°