How many practice questions are available for Circle Theorems?

SparkEd Math offers 60+ practice questions for Circle Theorems across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Circle Theorems content free on SparkEd Math?

Yes! All Circle Theorems content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Circle Theorems differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Circle Theorems problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Circle Theorems content aligned with the IB syllabus?

Yes, all Circle Theorems questions and lessons on SparkEd Math are fully aligned with the International Baccalaureate (MYP) (IB) syllabus for Class 10. The content follows the official curriculum structure and exam pattern.

Circle Theorems — Class 10 IB (2025-26)

About Circle Theorems — Class 10 IB

Prove and apply circle theorems including angle in a semicircle, cyclic quadrilaterals, and tangents. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 5). On this page you can practice 58 questions across three difficulty levels — 20 easy, 18 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Circle Theorems — solved examples for Class 10 IB

Example 1easy

Consider a circle with diameter AB. If C is any point on the circumference of the circle (distinct from A and B), which of the following statements about angle ACB is true?

A)Angle ACB is always acute.
B)Angle ACB is always obtuse.
C)Angle ACB is always 90°.
D)Angle ACB varies depending on the position of C.

Step-by-step solution

The diameter of a circle subtends an angle at any point on the circumference.
According to the angle in a semicircle theorem, this angle is always a right angle, i.e., 90°.

Answer: Angle ACB is always 90°.

Example 2medium

In a circle with centre O, A, B, and C are points on the circumference. If the angle subtended by the arc AC at the centre is ∠AOC = 110°, what is the measure of the angle subtended by the same arc at point B on the remaining part of the circle, i.e., ∠ABC?

A)50°
B)55°
C)110°
D)220°

Step-by-step solution

According to the Circle Theorem, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Given ∠AOC (angle at the centre) = 110°.
Therefore, ∠ABC (angle at the circumference) = (1/2) × ∠AOC.
∠ABC = (1/2) × 110° = 55°.

Answer: 55°

Example 3hard

In a circle with centre O, A, B, C, D are four points on the circumference such that ABCD is a cyclic quadrilateral. If the reflex angle BOD is 250°, what is the measure of angle BCD?

A)125°
B)115°
C)105°
D)135°

Step-by-step solution

The angle subtended by arc BCD at the centre (minor angle BOD) is 360° - reflex ∠BOD.
$Minor ∠BOD = 360° - 250° = 110°$
The angle subtended by arc BCD at the circumference is ∠BAD. This is half the minor central angle.
$∠BAD = (1/2) × Minor ∠BOD = (1/2) × 110° = 55°$
Since ABCD is a cyclic quadrilateral, the sum of opposite angles is 180°.
$∠BAD + ∠BCD = 180°$
Substitute the value of ∠BAD to find ∠BCD.
$∠BCD = 180° - 55° = 125°$

Answer: 125°

Practice questions on Circle Theorems

Q1.easy
In a circle with center O, points P, Q, and R are on the circumference. If ∠POQ = 110°, what is the measure of ∠PRQ? Assume R is on the major arc PQ.
1. A)55°
2. B)110°
3. C)220°
4. D)70°
Show answer
Answer: 55°
Hint: Relate the angle subtended by an arc at the center to the angle subtended by the same arc at any point on the remaining part of the circumference.
Q2.easy
Ravi was solving a problem involving a cyclic quadrilateral ABCD where ∠A = 70° and ∠B = 100°. He concluded that ∠C = 70° and ∠D = 100°. What is the mistake in Ravi's reasoning?
1. A)The sum of opposite angles in a cyclic quadrilateral is 360°, not 180°.
2. B)The sum of all angles in a cyclic quadrilateral is 180°.
3. C)Opposite angles in a cyclic quadrilateral are supplementary, meaning they sum to 180°.
4. D)Adjacent angles in a cyclic quadrilateral are always equal.
Show answer
Answer: Opposite angles in a cyclic quadrilateral are supplementary, meaning they sum to 180°.
Hint: Consider the property of opposite angles in a cyclic quadrilateral.
Q3.easy
In a circle, points A, B, C, and D lie on the circumference. If chords AC and BD intersect at a point E inside the circle, which statement is always true?
1. A)∠BAC = ∠BDC
2. B)∠ABC = ∠ADC
3. C)∠AEB = ∠CED
4. D)∠ACD = ∠ABD
Show answer
Answer: ∠ACD = ∠ABD
Hint: Look for angles subtended by the same arc in the same segment.
Q4.medium
Points P, Q, R, and S lie on a circle. If chord PQ subtends an angle ∠PRQ = 40° at point R on the circumference, what is the measure of the angle ∠PSQ subtended by the same chord PQ at point S on the circumference?
1. A)40°
2. B)80°
3. C)20°
4. D)100°
Show answer
Answer: 40°
Hint: Consider the theorem about angles subtended by the same segment of a circle.
Q5.medium
A diameter AB of a circle is 13 cm long. Point C is on the circumference such that the length of the chord AC is 5 cm. What is the length of the chord BC?
1. A)8 cm
2. B)12 cm
3. C)10 cm
4. D)14 cm
Show answer
Answer: 12 cm
Hint: Think about the special property of an angle inscribed in a semicircle and how it relates to triangles.
Q6.medium
ABCD is a cyclic quadrilateral. If the measure of angle ∠A is 70°, what is the measure of angle ∠C?
1. A)70°
2. B)90°
3. C)100°
4. D)110°
Show answer
Answer: 110°
Hint: Recall the property of opposite angles in a cyclic quadrilateral.
Q7.hard
From an external point P, two tangents PA and PB are drawn to a circle with centre O. A chord AB is drawn. Which of the following statements about triangle PAB is *always* true?
1. A)It is an equilateral triangle.
2. B)It is an isosceles triangle with PA = PB.
3. C)It is a right-angled triangle at P.
4. D)The line segment OP bisects ∠PAB.
Show answer
Answer: It is an isosceles triangle with PA = PB.
Hint: Recall the fundamental properties of tangents drawn from an external point to a circle, specifically regarding their lengths.
Q8.hard
A tangent AB touches a circle at point P. Chord PQ is drawn. Point R is on the circumference such that PR = RQ. If angle BPQ = 60°, what is the measure of angle PRQ?
1. A)30°
2. B)45°
3. C)60°
4. D)75°
Show answer
Answer: 60°
Hint: Apply the Alternate Segment Theorem carefully. Then use the property of isosceles triangles and the angle sum property of triangles.
Q9.hard
In a circle, AC is a diameter. Point B is on the circumference. Point D lies on the diameter AC such that BD is perpendicular to AC. If angle BAC = 35°, what is the measure of angle CBD?
1. A)25°
2. B)30°
3. C)45°
4. D)35°
Show answer
Answer: 35°
Hint: Use the property of the angle in a semicircle. Also, consider the angles in the right-angled triangles formed.

These are 9 of the 58 questions available for Circle Theorems. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.