Circles Class 10 ICSE — A. Das Gupta Chapter 20
Circles class 10 ICSE mathematics, chapter 20 from A. Das Gupta’s textbook, explains the angle properties of a circle through seven theorems and four arc properties. This chapter shows how angles at the centre relate to angles at the circumference, how angles in the same segment behave, and how cyclic quadrilaterals work. Each theorem builds on the one before it, and the exercises test both calculation and proof.
Download Angle Properties of a Circle Class 10 ICSE Mathematics Chapter 20 PDF
Download the Angle Properties of a Circle Class 10 ICSE Mathematics chapter PDF from the A. Das Gupta textbook to revise offline or print for practice. It is the full chapter as printed in the book — 27 pages, 2 sections, 35 questions.
What is inside Chapter 20
27 pages · 2 sections · 35 exercise questions
| Section | Title |
|---|---|
| 1 | Some Important Theorems. |
| 2 | Arc Properties of Circles (Theorems) |
- EXERCISE – 20 (A): 26 questions
- EXERCISE 20 (B): 9 questions
1 Some Important Theorems.
The chapter opens with its most used result. Theorem 1 says the angle an arc subtends at the centre is double the angle it subtends at any point on the remaining part of the circle: \angle AOB = 2 \angle ACB.
The proof joins C to O and extends it to D. In triangle AOC, OA = OC, so the base angles are equal. The exterior angle AOD equals the sum of the interior opposite angles, giving 2 \angle OCA, and the other side gives 2 \angle OCB. The proof runs three cases: C on the minor arc, on the major arc, and the difference case.
On the major arc, the central angle used is the reflex angle.
Theorem 2 follows directly: angles in the same segment are equal. Both \angle ACB and \angle ADB stand on arc AB, each half of \angle AOB, so \angle ACB = \angle ADB.
Theorem 3 states that the angle in a semicircle is a right angle. The arc is a straight line, so the central angle is 180° and half of it is 90°. Theorem 4 is the converse: if an arc subtends 90° at a point on the remaining part, the arc is a semicircle.
Theorem 5 says opposite angles of a cyclic quadrilateral are supplementary: \angle ADC + \angle ABC = 180^\circ. The proof adds the two central angles around O, which total 360°, then halves them. Theorem 6 is the converse: if a pair of opposite angles is supplementary, the quadrilateral is cyclic. Theorem 7 states that the exterior angle of a cyclic quadrilateral equals its interior opposite angle.
Worked pattern: in a circle with centre O, let \angle AOB = 76^\circ, with C on the major arc AB. Then \angle ACB = \frac{1}{2} \times 76^\circ = 38^\circ.
Common mistake: use the reflex central angle when the point lies on the major arc. With \angle AOC = 130^\circ, the reflex angle is 360^\circ - 130^\circ = 230^\circ, so the angle at the circumference is 115^\circ, not 65°.
2 Arc Properties of Circles (Theorems)
This section links arcs to angles and chords. Theorem 1 states that in equal circles — or the same circle — if two arcs subtend equal angles at the centre, the arcs are equal. The proof uses superposition: place one circle on the other so the centres and two radii align. Equal radii and equal central angles force the second radius onto the matching one, so the arcs coincide.
Theorem 2 is its converse: equal arcs subtend equal angles at the centre. Placing the arcs on top of each other makes the two sectors coincide, so the central angles match.
Theorem 3 states that equal chords cut off equal arcs. The proof splits into three cases. For minor arcs, triangles OAB and O’CD are congruent by SSS, so the central angles are equal, which gives equal arcs. For major arcs, the result reduces to the minor-arc case. For diameters, both arcs are semicircles and equal.
Theorem 4 is the converse: equal arcs have equal chords. For minor arcs, equal arcs give equal central angles, and the triangles match by SAS, so the chords are equal. The major-arc and semicircle cases follow the same reduction.
Worked pattern: take two equal circles, each with a central angle of 52^\circ. By Theorem 1, the arcs cut off are equal: \widehat{AB} = \widehat{CD}. Proof questions in Exercise 20 (B) use this link — equal arcs create equal angles at the circumference. For more practice on circles and constructions, see the Class 10 Maths book hub.
| Case | How the theorem handles it |
|---|---|
| Minor arc | Direct proof: SSS congruence in Theorem 3, SAS in Theorem 4 |
| Major arc | Reduced to the minor arc of the same chord |
| Semicircle | Chord is a diameter; semicircles of equal circles are equal |
What to practise from Circles Class 10 ICSE
The chapter closes with two exercise blocks. EXERCISE – 20 (A) has 26 questions on angle calculation. They apply the theorems from Section 1: central angle, same segment, semicircle, and cyclic quadrilateral.
EXERCISE 20 (B) has 9 questions, all proofs. They use the arc properties from Section 2: equal arcs, equal chords, and equal angles at the centre.
This split is the chapter’s exam pattern: one block for finding angles, one for proving statements about arcs and chords. Across the circles class 10 icse exercises, the same theorems recur. For other chapters, browse the Class 10 books hub. For the current syllabus and exam updates, check the official CISCE site.
Circles Class 10 ICSE frequently asked questions
What are the main theorems in circles class 10 ICSE?
The chapter has seven angle theorems and four arc properties. The core results are: the angle at the centre is double the angle at the circumference, angles in the same segment are equal, the angle in a semicircle is 90°, and opposite angles of a cyclic quadrilateral sum to 180°.
Why is the angle in a semicircle always a right angle?
The arc is a straight line through the centre, so the central angle is 180°. The angle at the circumference is half of that, which gives 90°.
How do I know whether to use the central angle or its reflex angle?
Look at which arc the point sits on. If the point is on the major arc, the relevant central angle is the reflex one. Subtract the given small angle from 360° before halving it.
What does the exterior angle property of a cyclic quadrilateral say?
When a side of a cyclic quadrilateral is extended, the exterior angle formed equals the interior opposite angle. This is Theorem 7 of the chapter.
Where can I find the circles class 10 ICSE chapter PDF?
The chapter PDF is available on this page. Use the download block above to access the A. Das Gupta chapter 20 material for offline study.
Reference: ICSE Class 10 Mathematics textbook material, chapter 20: Angle Properties of a Circle — CISCE-aligned.
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