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# 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 △*OBC*is 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 =2*θ

But, recall the inscribed angle theorem.

∠*COB = 2*∠*BAC*

*2*θ = 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°.