This article covers the essential skills of circle properties – a core topic in DSE Paper 1 Section A(2) and Section B worth 4–7 marks. You will learn the basic definitions of circle elements (centre, radius, chord, arc, sector), the relationships between central angles, arcs, and chords, the inscribed angle theorem, the properties of cyclic quadrilaterals, and how to apply these theorems in geometric proofs. The article includes step-by-step worked examples, DSE exam techniques, practice questions. These are essential skills that appear regularly in DSE papers.
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Circle Properties (Part 1) | Angles,Arcs & Chords
Question 1 MC
In a circle, arc \( AB \) measures \( 120^\circ \). What is the measure of the central angle \( \angle AOB \)?
A. \( 60^\circ \) B. \( 120^\circ \) C. \( 240^\circ \) D. \( 180^\circ \)
Question 2 MC
In the figure, \( O \) is the centre. \( \angle AOB = 140^\circ \). Find \( \angle ACB \).
A. \( 70^\circ \) B. \( 140^\circ \) C. \( 40^\circ \) D. \( 110^\circ \)
Question 3 Short Answer
In cyclic quadrilateral \( ABCD \), \( \angle B = 110^\circ \). Find \( \angle D \).
Question 4 Short Answer
\( AB \) is a diameter. \( C \) is on the circle. If \( \angle ABC = 44^\circ \), find \( \angle BAC \).
Question 5 MC
In the figure, \( ABCD \) is cyclic. \( \angle A = 2x \), \( \angle C = 3x - 20^\circ \). Find \( x \).
A. \( 30^\circ \) B. \( 40^\circ \) C. \( 50^\circ \) D. \( 60^\circ \)
Question 6 Short Answer
In a circle, chords \( AB = CD \). What can you conclude about the arcs \( AB \) and \( CD \)?
Question 7 MC
An inscribed angle intercepts an arc of \( 110^\circ \). What is the measure of the angle?
A. \( 55^\circ \) B. \( 110^\circ \) C. \( 220^\circ \) D. \( 70^\circ \)
Question 8 Short Answer
Prove that opposite angles of a cyclic quadrilateral are supplementary.
Question 9 Short Answer
In the figure, \( O \) is the centre. \( \angle AOB = 90^\circ \). Find \( \angle ACB \).
Question 10 Short Answer
In a cyclic quadrilateral \( ABCD \), \( \angle A = 75^\circ \), \( \angle B = 100^\circ \). Find \( \angle C \) and \( \angle D \).
When using the inscribed angle theorem, remember that the inscribed angle is half the intercepted arc, not equal to it. Also, the angle at the centre is twice the inscribed angle on the same arc.