# Alternate Segment Theorem – Explanation & Examples

There exist several geometric properties and theorems about circles. Circle theorems are very useful because they are used in geometric proofs and to calculate angles.

You have studied the Inscribed Angle Theorem and Thales’ Theorem so far. In this article, you will learn about an interesting theorem known as the Alternate Segment Theorem. Like the other two theorems, this is also based on the angles.

## What is the Alternate Segment Theorem?

The alternate segment theorem also referred to as the tangent-chord theorem, states that:

The angle measure between a chord of a circle and a tangent through any of the chord’s endpoints is equal to the measure of an angle in the alternate segment.

According to the alternate segment theorem, ∠CBD = ∠CAB

α = θ

Where α and θ are alternate angles.

Proof of alternate segment theorem:

Let’s get a clear understanding of the theorem by making a few proofs.

• Join the ends of all the cords to the center of the circle. These will be the radii of the circle.
• Since, OB = OA = OC, then △OBCis isosceles, so we have

OCB =OBC

COB = 180°− ∠OCB − ∠OBC

= 180° − 2∠OBC ………………………(i)

• Since OB (radius) joins the tangent BD at point B, then ∠OBD = 90°

Therefore, θ = 90°− ∠OBC…………………. (ii)

By solving equation (i) and (ii), we get

COB =2θ

But, recall the inscribed angle theorem.

COB = 2BAC

2θ = 2∠BAC

Divide both sides by 2 to get,

BAC = θ

For a better understanding of the theorem, let’s work through some examples:

Example 1

Find the value of ∠QPS in the diagram shown below.

Solution

By alternate segment theorem,

QPS = QRP

So, ∠QPS = 70°

Example 2

In the diagram below, ∠CBD = 56° and ∠ABC = 65°. What is the measure of ∠ACB?

Solution

Alternate segment theorem tells us that,

CBD =∠BAC = 56°

And according to triangle sum theorem,

ABC + ACB + BAC = 180°

65° + ∠ACB + 56° = 180°

Simplify.

121° + ∠ACB = 180°

Subtract 121° on both sides.

ACB = 59°

Therefore, the measure of ∠ACB is 59°.

Example 3

In the diagram shown below, point C is the center of the circle with a radius of 8 cm and ∠QRS = 80°. Find the length of arc QTR.

Solution

First, join the vertices of the triangle to the center.

By alternate segment theorem, ∠QRS=QPR = 80°.

Recall the inscribed angle theorem, 2∠QPR = QCR.

So, ∠QCR = 2 x 80°.

= 160°.

Arc length = 2πr(θ/360)

= 2 x 3.14 x 8 x (160/360)

= 22.33 cm.

Example 4

In the diagram below, point C is the center of the circle. If ∠AEG = 160° and ∠DEF =60°, find the measure of ∠EAB and ∠ BDE

Solution

According to the tangent-chord theorem,

EAB = DEF =60°

Similarly,

AEG = BDE = 160°

Example 5

Find the measure of angle x and y in the diagram below.

Solution

Length AB = BC (property of tangents)

COA = 180° – (90 + 35°/2)

= 160° – 107.5°

= 72.5°

Therefore, ∠ AOB = 2 x 72.5°

= 145°

Recalling the inscribed angle theorem,

2x = ∠ AOB = 145°

x = 72.5°.

And by alternate segment theorem,

x = y = 72.5°

Example 6

In the diagram below, AB is the diameter of the circle. Find the measure of angles x, y, and z.

Solution

According to the inscribed angle theorem, z = 90°

And,

sum of interior angles of a triangle =180°

So, x = 180° – (90° + 18°)

x = 72°

Also, according to alternate segment theorem,

x = y = 72°

Therefore, the measure of angle x = y =72° and z = 90°

Example 7

Find the measure of ∠x and ∠y in the diagram below.

Solution

Sum of interior angles of a triangle = 180°.

50° + 50° + x = 180°

x = 180° – 100°

x = 80°

And according to alternate segment theorem,

x = y = 80°.

Therefore, the measure of ∠x and ∠y is 80°.

Example 8

Given ABC is 70 degrees and angle BCD is 66 degrees. What is the measure of angle x?

Solution

Angle BCD = angle CAB = 66° (Alternate segment theorem).

And sum of interior angles = 180°

70° + 66° + x = 180°

Simplify.

136° + x = 180°

Subtract 136° on both sides.

x = 44°.

Thus, the measure of angle x is 44°.

### Practice Questions

1. True or False: In the alternate segment theorem, if a triangle is inscribed in a circle, a tangent at any of the three points of intersections of a circle and a triangle will make angles equal to that in the alternate segment.

2. True or False: In the alternate segment theorem, the angle between the chord and the tangent is not equal to the angle in the alternate segment.

3. Fill in the blank: The angle that is made in another sector from a chord is called __________.

4. Fill in the blank: The angle made at the center of the circle is __________, the value of the angle made at the circumference by the same arc.

5. In the diagram below, $\angle CBD = 48^{\circ}$ and $\angle ABC = 56^{\circ}$. What is the measure of $\angle ACB$?