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:

$\angle TAB=\angle ACB\text{or}\angle a=\angle b$

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:

$$\begin{gathered} OA\perp TP\text{(Radius is perpendicular to the tangent)} \\ a+x=90^{\circ }\dots \dots .(1) \\ \text{In }△\text{OAD, right angled at D.} \\ 90^{\circ }+x+y=180^{\circ } \\ \Rightarrow x+y=180^{\circ }-90^{\circ } \\ \Rightarrow x+y=90^{\circ }\dots \dots (2) \\ \text{Putting the value of x from (1) in (2)} \\ 90^{\circ }-a+y=90^{\circ } \\ \Rightarrow y=a\dots \dots \dots .(3) \\ \text{By Theorem: The angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle.} \\ \angle AOB=\angle ACB \\ \Rightarrow 2y=2b \\ \Rightarrow y=b \\ \text{From (3),}a=b \\ \Rightarrow \angle TAB=\angle ACB \\ Hence,\angle \text{ between the tangent and chord }=\text{ angle in alternate segment} \end{gathered}$$

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:

$\angle TAB=\angle ACB=40^{\circ }$

Q2. In a circle, the angle between a tangent and a chord is $65^{\circ }$. Find the angle in the alternate segment.

Answer:  

$\angle \text{in alternate segment}=65^{\circ }$

Q3. A tangent touches a circle at point A. A chord AB is drawn. If $\angle ACB=50^{\circ }$, what is $\angle TAB$?

Answer:

$\angle TAB=\angle ACB=50^{\circ }$

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:

$$\begin{gathered} \text{By the Alternate Segment Theorem:} \\ \angle ACB=\angle \text{ between tangent and chord }=40^{\circ } \\ \text{Answer:}\angle ACB=40^{\circ } \end{gathered}$$

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:

$$\begin{gathered} \text{By the Alternate Segment Theorem:} \\ \angle TAB=\angle ACB=50^{\circ } \\ \text{Answer:}\angle TAB=50^{\circ } \end{gathered}$$

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

Solution:

$$\begin{gathered} \text{In triangle ABC:} \\ \angle ABC=130^{\circ } \\ \Rightarrow \angle ACB=180^{\circ }-\angle ABC-\angle BAC \\ \text{But since}(\angle TAB=\angle ACB)(AlternateSegmentTheorem), \\ \text{Let’s use directly:} \\ \angle TAB=\angle ACB=180^{\circ }-130^{\circ }=50^{\circ } \\ \text{Answer:}(\angle TAB=50^{\circ }) \end{gathered}$$

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

Solution:

$$\begin{gathered} \text{In a circle, a tangent touches point A. Chords AB and AC are drawn. If}\angle TAB=30^{\circ }\text{and}\angle TAC=30^{\circ },\text{ prove that }\angle ABC=\angle ACB. \\ \text{Solution:} \\ ByAlternateSegmentTheorem: \\ (\angle ABC=\angle TAB=30^{\circ }) \\ (\angle ACB=\angle TAC=30^{\circ }) \\ Thus,\angle ABC=\angle ACB \\ \text{Proved} \end{gathered}$$

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 $\angle BCA.$

Solution:

$$\begin{gathered} \text{Solution:} \\ \text{By the Alternate Segment Theorem:} \\ \angle BCA=\angle TAB=35^{\circ } \\ \text{Answer:}(\angle BCA=35^{\circ }) \end{gathered}$$

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 $42^{\circ }$, find the angle in the alternate segment subtended by the chord.

Question 2: In a circle, the angle $\angle ABC=56^{\circ }$, 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 $60^{\circ }$. If point CC lies in the alternate segment, find the measure of $\angle ACB$.

Question 4: The chord AB subtends an angle $\angle ACB$ in the alternate segment of the circle. If $\angle ACB=35^{\circ }$, 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 $\angle ABC=40^{\circ }$. Use the Alternate Segment Theorem to find the value of $\angle TAB$.

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 $\angle TAB=x$, 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 $\angle ACB=50^{\circ }$, find the angle between the tangent and chord AB.