How do you prove that four points lie on one circle?

If two angles standing on the same side are equal, the four points are concyclic; alternatively show opposite angles of the quadrilateral sum to 180.

Why can an angle at the circumference never reach 180 degrees?

The arc it stands on must be less than the whole circle, so the central angle is below 360 and half of it stays below 180.

S5 Maths: Circle Geometry Paper 4 (concyclic points, arc ratios and combined angles)

2026-11-03

S · 5

This paper reverses the earlier theorems: equal angles at the circumference prove four points concyclic, a 90 degree angle identifies a diameter, and tangent lengths combine with Pythagoras. Arc ratios are always turned into central angles first, then halved for the circumference angle. The last two items use the intersecting-chords and secant formulas in reverse. Every value is checked by computer.

Quick answer　Equal angles at the circumference imply four concyclic points; a diameter of 10 has radius 5; with OP 13 and r 5 the tangent is 12; a circumference angle of 30 gives a central angle of 60, one sixth of the turn; arcs in ratio 3 to 7 give 108 and 252, and 54 and 126 at the circumference; tangent angle 70 gives base angles of 55; exterior angle 118 leaves an adjacent 62; an interior angle of 80 implies the other arc is 40; tangents at 60 degrees cut arcs of 240 and 120; an angle at the circumference can never be 180.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	25 minutes
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: Circle Geometry
Level	S5

02　Questions (10 items)

FoundationFor four points A, B, C, D the angles ACB and ADB are on the same side and equal. What follows?
FoundationA circle has a diameter of 10 cm. What is the radius, and what is the largest angle at the circumference?
FoundationP is outside the circle, PT is a tangent at T, OP is 13 cm and the radius is 5 cm. Find PT.
FoundationAn arc subtends 30 degrees at the circumference. Find the central angle and the arc fraction of the whole circle.
IntermediateTwo arcs are in the ratio 3 to 7 and together make the whole circle. Find their central angles and the angles they subtend at the circumference.
IntermediatePA and PB are tangents with angle APB equal to 70 degrees. Find angle PAB and angle AOB.
IntermediateAn exterior angle of a cyclic quadrilateral is 118 degrees. Find the interior opposite and adjacent angles.
ChallengeTwo chords intersect inside the circle at 80 degrees and one intercepted arc is 120 degrees. Find the other arc.
ChallengeTwo tangents from an external point meet at 60 degrees. Find the major and minor arcs between the points of contact.
Error redoA student says an angle at the circumference can reach 180 degrees if the arc is large enough. What is wrong?

03　Answers and one-line notes

#	Answer	One-line note
1	The four points are concyclic.	Equal angles on the same side stand on the same arc, so the four points lie on one circle.
2	Radius 5 cm, and the largest is 90 degrees, excluding the limit.	The radius is half the diameter, and the arc can approach a semicircle, so the bound is 90.
3	12 cm.	The tangent is perpendicular to the radius, so 13 squared minus 5 squared gives 144 and the root is 12.
4	Central angle 60 degrees, one sixth of the circle.	The central angle is 60, and 60 divided by 360 gives one sixth.
5	Central angles 108 and 252, with 54 and 126 at the circumference.	360 times three tenths gives 108 and times seven tenths gives 252; halving gives 54 and 126.
6	Angle PAB is 55 and angle AOB is 110 degrees.	The isosceles base angle is (180 minus 70) over 2, giving 55; with two right angles at the centre, angle AOB is 180 minus 70, giving 110.
7	Interior opposite 118 and adjacent 62 degrees.	The exterior angle equals the interior opposite angle, and the adjacent angle on the straight line is 180 minus 118, giving 62.
8	40 degrees.	The interior angle is half the sum of the arcs, so the arcs total 160 and 160 minus 120 gives 40.
9	Major arc 240 and minor arc 120 degrees.	The angle is half the difference, so the difference is 120; with a total of 360 the arcs are 240 and 120.
10	The arc can never be the whole circle, so the angle at the circumference is always below 180 degrees.	The angle at the circumference is half the central angle, which is under 360, so half stays under 180.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.