What is the alternate segment in a circle?

The alternate segment is the region of a circle opposite the angle formed between a chord and the tangent at the point of contact. It’s where the angle corresponding to the tangent-chord angle lies.

What are some common alternate segment theorem questions in exams?

Typical alternate segment theorem questions involve finding unknown angles when a tangent and chord are given. For example: "If the angle between a tangent and a chord is 40°, find the angle in the alternate segment."

What is the proof of the Alternate Segment Theorem?

The alternate segment theorem proof involves using: The perpendicularity of the radius to the tangent Properties of cyclic quadrilaterals Basic angle chasing in triangles formed inside the circle It shows that the angle between the tangent and the chord equals the angle in the opposite (alternate) segment.

Is the Alternate Segment Theorem applicable to all circles?

Yes, the theorem applies to all circles where a tangent and chord are drawn from the same point of contact. The only requirement is that the tangent must touch the circle at one point, and the chord must pass through that point.

Can the Alternate Segment Theorem be used in coordinate geometry?

While the alternate segment theorem is a geometric concept, it can sometimes be used in coordinate geometry when problems involve tangent lines, chords, and angle calculations in a coordinate plane.

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Alternate Segment Theorem

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Alternate Segment Theorem

The Alternate Segment Theorem is a key result in circle geometry that connects tangents and chords. It states that the angle between a tangent and a chord through the point of contact is equal to the angle formed in the alternate segment of the circle. This theorem is widely used to solve problems involving angles in circles and appears frequently in exams. Understanding this concept helps in mastering circle theorems and their applications in geometry.

1.0What is an Alternate Segment?

In a circle, when a tangent touches the circle and a chord is drawn from the point of contact, the circle is divided into two segments. The part of the circle that lies opposite the angle formed between the chord and the tangent is called the alternate segment.

2.0Alternate Segment Theorem Statement

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle.

This theorem plays a key role in solving problems involving angles in a circle.

3.0Alternate Segment Theorem Proof

Let’s understand the proof of the alternate segment theorem step-by-step:

Given:

A circle with center O
A tangent TP touching the circle at point A
A chord AB drawn from the point of contact
A point C on the circle such that _{{\angle AC}} lies in the alternate segment

To Prove:

$∠ T A B = ∠ A CB or ∠ a = ∠ b$

Construction:

Join OA and OB, the radii of the circle, and form triangle OAB. Then, draw a perpendicular from the center O to the chord AB, and mark the foot of the perpendicular as D.

Proof:

$O A ⊥ TP (Radius is perpendicular to the tangent) a + x = 9 0^{\circ} \dots\dots . (1) In △ OAD, right angled at D. 9 0^{\circ} + x + y = 18 0^{\circ} \Rightarrow x + y = 18 0^{\circ} - 9 0^{\circ} \Rightarrow x + y = 9 0^{\circ} \dots\dots (2) Putting the value of x from (1) in (2) 9 0^{\circ} - a + y = 9 0^{\circ} \Rightarrow y = a \dots\dots\dots . (3) By Theorem: The angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle. ∠ A OB = ∠ A CB \Rightarrow 2 y = 2 b \Rightarrow y = b From (3), a = b \Rightarrow ∠ T A B = ∠ A CB He n ce, ∠ between the tangent and chord = angle in alternate segment$

4.0Alternate Segment Theorem Questions

Here are a few typical questions that apply the theorem:

Q1. In the given circle, AB is a chord and TP is a tangent at point A. If _{\angle TAB = 40^\circ}, find _{\angle ACB}.

Solution:
Using the alternate segment theorem:

$∠ T A B = ∠ A CB = 4 0^{\circ}$

Q2. In a circle, the angle between a tangent and a chord is $6 5^{\circ}$ . Find the angle in the alternate segment.

Answer:

$∠ in alternate segment = 6 5^{\circ}$

Q3. A tangent touches a circle at point A. A chord AB is drawn. If $∠ A CB = 5 0^{\circ}$ , what is $∠ T A B$ ?

Answer:

$∠ T A B = ∠ A CB = 5 0^{\circ}$

5.0Applications of the Alternate Segment Theorem

Solving problems involving angles in circles
Finding unknown angles when tangents and chords are involved
Frequently used in geometry proofs and construction problems
Appears in exams like JEE, CBSE Board, and Olympiads

6.0Solved Examples on Alternate Segment Theorem

Example 1: In a circle, a tangent touches the circle at point A. A chord AB is drawn. If the angle between the tangent and chord is _{40^\circ}, find _{{\angle ACB}}, where C lies in the alternate segment.

Solution:

$By the Alternate Segment Theorem: ∠ A CB = ∠ between tangent and chord = 4 0^{\circ} Answer: ∠ A CB = 4 0^{\circ}$

Example 2: In the figure, chord ABAB subtends an angle of _{50^\circ} at point C in the alternate segment. Find the angle between the tangent at point A and chord AB.

Solution:

$By the Alternate Segment Theorem: ∠ T A B = ∠ A CB = 5 0^{\circ} Answer: ∠ T A B = 5 0^{\circ}$

Example 3: In a circle, a tangent at point A and chord AB are drawn. If , find .

Solution:

$In triangle ABC: ∠ A BC = 13 0^{\circ} \Rightarrow ∠ A CB = 18 0^{\circ} - ∠ A BC - ∠ B A C But since (∠ T A B = ∠ A CB) (A lt er na t e S e g m e n tT h eore m), Let’s use directly: ∠ T A B = ∠ A CB = 18 0^{\circ} - 13 0^{\circ} = 5 0^{\circ} Answer: (∠ T A B = 5 0^{\circ})$

Example 4: In a circle, a tangent touches point A. Chords AB and AC are drawn. If and , prove that .

Solution:

$In a circle, a tangent touches point A. Chords AB and AC are drawn. If ∠ T A B = 3 0^{\circ} and ∠ T A C = 3 0^{\circ}, prove that ∠ A BC = ∠ A CB . Solution: B y A lt er na t e S e g m e n tT h eore m : (∠ A BC = ∠ T A B = 3 0^{\circ}) (∠ A CB = ∠ T A C = 3 0^{\circ}) T h u s, ∠ A BC = ∠ A CB Proved$

Example 5: In a circle, a tangent at point A and chord AB are drawn. The angle between the tangent and chord is _{{35^\circ }}. Point C lies in the alternate segment. Find $∠ BC A .$

Solution:

$Solution: By the Alternate Segment Theorem: ∠ BC A = ∠ T A B = 3 5^{\circ} Answer: (∠ BC A = 3 5^{\circ})$

7.0Practice Questions on Alternate Segment Theorem

Question 1: A tangent touches a circle at point A, and a chord AB is drawn. If the angle between the tangent and chord is $4 2^{\circ}$ , find the angle in the alternate segment subtended by the chord.

Question 2: In a circle, the angle $∠ A BC = 5 6^{\circ}$ , where point C lies in the alternate segment. Find the angle between the tangent at point A and the chord AB.

Question 3: In a circle with center O, the tangent at point A and the chord AB form an angle of $6 0^{\circ}$ . If point CC lies in the alternate segment, find the measure of $∠ A CB$ .

Question 4: The chord AB subtends an angle $∠ A CB$ in the alternate segment of the circle. If $∠ A CB = 3 5^{\circ}$ , find the angle between the chord AB and the tangent at point A.

Question 5: In the figure, a tangent touches the circle at point A, and a chord AB is drawn $∠ A BC = 4 0^{\circ}$ . Use the Alternate Segment Theorem to find the value of $∠ T A B$ .

Question 6 : If a circle has two chords AB and AC, and both make the same angle with the tangent at point A, prove that triangle ABC is isosceles.

Question 7: A circle has a tangent at point A and a chord AB. If $∠ T A B = x$ , and the angle in the alternate segment is 2x, find the value of x.

Question 8: In a triangle ABC, AB is a chord of a circle, and the circle touches the triangle at point A. If $∠ A CB = 5 0^{\circ}$ , find the angle between the tangent and chord AB.

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(Session 2026 - 27)

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Class

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Preferred Programs

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Alternate Segment Theorem

The Alternate Segment Theorem is a key result in circle geometry that connects tangents and chords. It states that the angle between a tangent and a chord through the point of contact is equal to the angle formed in the alternate segment of the circle. This theorem is widely used to solve problems involving angles in circles and appears frequently in exams. Understanding this concept helps in mastering circle theorems and their applications in geometry.

1.0What is an Alternate Segment?

In a circle, when a tangent touches the circle and a chord is drawn from the point of contact, the circle is divided into two segments. The part of the circle that lies opposite the angle formed between the chord and the tangent is called the alternate segment.

2.0Alternate Segment Theorem Statement

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle.

This theorem plays a key role in solving problems involving angles in a circle.

3.0Alternate Segment Theorem Proof

Let’s understand the proof of the alternate segment theorem step-by-step:

Given:

A circle with center O
A tangent TP touching the circle at point A
A chord AB drawn from the point of contact
A point C on the circle such that _{{\angle AC}} lies in the alternate segment

To Prove:

$∠ T A B = ∠ A CB or ∠ a = ∠ b$

Construction:

Join OA and OB, the radii of the circle, and form triangle OAB. Then, draw a perpendicular from the center O to the chord AB, and mark the foot of the perpendicular as D.

Proof:

$O A ⊥ TP (Radius is perpendicular to the tangent) a + x = 9 0^{\circ} \dots\dots . (1) In △ OAD, right angled at D. 9 0^{\circ} + x + y = 18 0^{\circ} \Rightarrow x + y = 18 0^{\circ} - 9 0^{\circ} \Rightarrow x + y = 9 0^{\circ} \dots\dots (2) Putting the value of x from (1) in (2) 9 0^{\circ} - a + y = 9 0^{\circ} \Rightarrow y = a \dots\dots\dots . (3) By Theorem: The angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle. ∠ A OB = ∠ A CB \Rightarrow 2 y = 2 b \Rightarrow y = b From (3), a = b \Rightarrow ∠ T A B = ∠ A CB He n ce, ∠ between the tangent and chord = angle in alternate segment$

4.0Alternate Segment Theorem Questions

Here are a few typical questions that apply the theorem:

Q1. In the given circle, AB is a chord and TP is a tangent at point A. If _{\angle TAB = 40^\circ}, find _{\angle ACB}.

Solution:
Using the alternate segment theorem:

$∠ T A B = ∠ A CB = 4 0^{\circ}$

Q2. In a circle, the angle between a tangent and a chord is $6 5^{\circ}$ . Find the angle in the alternate segment.

Answer:

$∠ in alternate segment = 6 5^{\circ}$

Q3. A tangent touches a circle at point A. A chord AB is drawn. If $∠ A CB = 5 0^{\circ}$ , what is $∠ T A B$ ?

Answer:

$∠ T A B = ∠ A CB = 5 0^{\circ}$

5.0Applications of the Alternate Segment Theorem

Solving problems involving angles in circles
Finding unknown angles when tangents and chords are involved
Frequently used in geometry proofs and construction problems
Appears in exams like JEE, CBSE Board, and Olympiads