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# The Inscribed Angle Theorem â€“ Explanation & Examples

The circular geometry is really vast. A circle consists of many parts and angles. These parts and angles are mutually supported by certain Theorems, e.g., t**he Inscribed Angle Theorem**, Thales’ Theorem, and Alternate Segment Theorem.

**We will go through the inscribed angle theorem**, but before that, letâ€™s have a brief overview of circles and their parts.

Circles are all around us in our world. There exists an interesting relationship among the angles of a circle. To recall, a chord of a circle is the straight line that joins two points on a circle’s circumference. Three types of angles are formed inside a circle when two chords meet at a common point known as a vertex. These angles are the central angle, intercepted arc, and the inscribed angle.

For more definitions related to circles, you need to go through the previous articles.

*In this article, you will learn:*

- The inscribed angle and inscribed angle theorem,
- we will also learn how to prove the inscribed angle theorem.

## Â

## What is the Inscribed Angle?

**An inscribed angle is an angle whose vertex lies on a circle, and its two sides are chords of the same circle.**

On the other hand, a central angle is an angle whose vertex lies at the center of a circle, and its two radii are the sides of the angle.

The intercepted arc is an angle formed by the ends of two chords on a circle’s circumference.

Letâ€™s take a look.

In the above illustration,

**Î±** = The central angle

**Î¸** = The inscribed angle

**Î²** = the intercepted arc.

## What is the Inscribed Angle Theorem?

*The inscribed angle theorem, which is also known as the arrow theorem or the central angle theorem, states that:*

**The size of the central angle is equal to twice the size of the inscribed angle. The inscribed angle theorem can also be stated as:**

**Î± = 2****Î¸**

The size of an inscribed angle is equal to half the size of the central angle.

**Î¸ = Â½****Î±**

Where Î± and Î¸ are the central angle and inscribed angle, respectively.

## How do you Prove the Inscribed Angle Theorem?

*The inscribed angle theorem can be proved by considering three cases, namely:*

- When the inscribed angle is between a chord and the diameter of a circle.
- The diameter is between the rays of the inscribed angle.
- The diameter is outside the rays of the inscribed angle.

**Case 1: When the inscribed angle is between a chord and the diameter of a circle:**

**To prove Î± = 2Î¸:**

- â–³
*CBD*is an isosceles triangle whereby*CD = CB*= the radius of the circle. - Therefore, âˆ CDB = âˆ DBC = inscribed angle = Î¸
- The diameter AD is a straight line, so âˆ
*BCD*= (180**â€“**Î±) Â° - By triangle sum theorem, âˆ
*CDB*+ âˆ DBC + âˆ BCD = 180Â°

Î¸ + Î¸ + (180 **â€“** Î±) = 180Â°

Simplify.

âŸ¹ Î¸ + Î¸ + 180 **â€“** Î± = 180Â°

âŸ¹ 2Î¸ + 180 â€“ Î± = 180Â°

Subtract 180 on both sides.

âŸ¹ 2Î¸ + 180 â€“ Î± = 180Â°

âŸ¹ 2Î¸ â€“ Î± = 0

**âŸ¹**** 2Î¸ = Î±. Hence proved.**

**Case 2: when the diameter is between the rays of the inscribed angle.**

To prove 2Î¸ = Î±:

- First, draw the diameter (in dotted line) of the circle.

- Let the diameter bisects Î¸ into Î¸
_{1}and Î¸ Similarly, the diameter bisects Î± into Î±_{1 }and Î±_{2}.

âŸ¹ Î¸_{1} + Î¸_{2} = Î¸

âŸ¹ Î±_{1 }+ Î±_{2} = Î±

- From the first case above, we already know that,

âŸ¹ 2Î¸_{1 }= Î±_{1}

âŸ¹ 2Î¸_{2} = Î±_{2}

- Add the angles.

âŸ¹ Î±_{1 }+ Î±_{2} = 2Î¸_{1 }+ 2Î¸_{2}

âŸ¹ Î±_{1 }+ Î±_{2} = 2 (Î¸_{1 }+ 2Î¸_{2})

**Hence, ****2Î¸ = Î±:**

**Case 3: When the diameter is outside the rays of the inscribed angle.**

To prove 2Î¸ = Î±:

- Draw the diameter (in dotted line) of the circle.

- Since 2Î¸
_{1}= Î±_{1}

âŸ¹ 2 (Î¸_{1 }+ Î¸) = Î± + Î±_{1}

âŸ¹ But, 2Î¸_{1 }= Î±_{1 }and 2Î¸_{2} = Î±_{2}

âŸ¹ By substitution, we get,

2Î¸ = Î±:

**Solved examples about inscribed angle theorem**

*Example 1*

Find the missing angle x in the diagram below.

__Solution__

By inscribed angle theorem,

The size of the central angle = 2 x the size of the inscribed angle.

Given, 60Â° = inscribed angle.

Substitute.

The size of the central angle = 2 x 60Â°

= 120Â°

*Example 2*

Given that âˆ *QRP* = (2x + 20) Â° and âˆ *PSQ *= 30Â°, find the value of x.

__Solution__

By inscribed angle theorem,

Central angle = 2 x inscribed angle.

âˆ *QRP =2*âˆ *PSQ*

âˆ *QRP *= 2 x 30Â°.

= 60Â°.

Now, solve for x.

âŸ¹ (2x + 20) Â° = 60Â°.

Simplify.

âŸ¹ 2x + 20Â° = 60Â°

Subtract 20Â° on both sides.

âŸ¹ 2x = 40Â°

Divide both sides by 2.

âŸ¹ x = 20Â°

So, the value of x is 20Â°.

*Example 3*

Solve for angle x in the diagram below.

__Solution__

Given the central angle = 56Â°

2âˆ *ADB =*âˆ *ACB*

2x = 56Â°

Divide both sides by 2.

x = 28Â°

*Example 4*

If âˆ *YMZ* = 150Â°, find the measure of âˆ *MZY* and âˆ *XMY.*

__Solution__

Triangle MZY is an isosceles triangle, Therefore,

âˆ *MZY =*âˆ *ZYM*

Sum of interior angles of a triangle = 180Â°

âˆ *MZY = *âˆ *ZYM = *(180Â° – 150Â°)/2

= 30Â° /2 = 15Â°

Hence, âˆ *MZY = *15Â°

And by inscribed angle theorem,

2âˆ *MZY = *âˆ *XMY*

âˆ *XMY *= 2 x 15Â°

= 30Â°