**Arcs
and Angles Subtended by Arcs on a Circle**

A portion (part) of the circumference of a circle is called an
**arc**. In the figure given below, ACB is a part of the circumference of the circle with
centre O. So ACB is an **arc**. There are minor **arcs** and major **arcs**. A minor **arc** is
less than the half of the circumference and major **arc** is greater than the half
of the circumference.

So, **arc ACB** is a minor arc and the **arc ADB** is a major arc.

**Central
angle and its corresponding arc**

In the figure given below, O is the centre of the circle in
which OP and OQ are two radii and PRQ is an arc. The arc PRQ subtends an angle
POQ at the centre O. So, ∠POR is called the **central
angle** and the **arc PRQ** is its corresponding arc.

Since both the central angle and corresponding arc are
described by the same amount of rotation, therefore the degree measure of arc
PRQ and ∠POQ
are same.

**i.e. ∠****POQ ≗ arc PRQ**

**Inscribed
angle and its corresponding arc**

In the figure given below, O is the centre of the circle in which AB and BC are two chords. The chords AB and BC meet at a point B on the circumference of the circle. So they form an angle ABC at B on the circumference.

The angle on the circumference is called the **inscribed angle**. So, ∠ABC
is an **inscribed angle** and **arc AMC** is its corresponding arc.

Now, to understand and establish the relation between an inscribed angle and
its corresponding arc, let us consider an example of the figure given below:

In DAOB,
∠OAB
= ∠OBA
[∵ OA = OB]

∠OAB
+ ∠OBA + ∠AOB = 180° [Sum of angles of a
triangle]

or, ∠OBA
+ ∠OBA + ∠AOB = 180°

or, 2∠OBA
= 180° - ∠AOB ………………. (i)

In DBOC,
∠OBC
= ∠OCB
[∵ OB = OC]

∠OBC
+ ∠OCB + ∠BOC = 180° [Sum of angles of a
triangle]

or, ∠OBC
+ ∠OBC + ∠BOC = 180°

or, 2∠OBC
= 180° - ∠BOC ………………. (ii)

Adding (i) and
(ii), we have

2∠OBA + 2∠OBC = 180° - ∠AOB + 180° - ∠BOC

or, 2(∠OBA
+ ∠OBC) = 360° - (∠AOB + ∠BOC)

or, 2∠ABC
= 360° - Reflex ∠AOC

or, 2∠ABC
≗ circumference – arc ABC [∵
Circumference ≗ 360° and Ref.∠AOC ≗ arc ABC]

or, 2∠ABC
≗ arc AMC

or, **∠ABC
≗**** ½ arc AMC**

**Thus, the
degree measurement of the angle on circumference of a circle is equal to the
half of the degree measurement of its corresponding arc. **

**Properties of circle related to
arcs and the angles subtended by them: **

*1. **Arcs
subtended by equal angles at the centre of the circle are equal.*

*2. **Angles
subtended by two equal arcs of a circle at the centre are equal.*

*3. **Arcs
cut off by equal chords of a circle are equal. Or, If two chords of a circle
are equal, the corresponding arcs are equal.*

*4. **If
two arcs of a circle are equal, then their corresponding chords are equal.*

*5. **Central
angle of a circle is equal to the degree measurement of its opposite arc.*

*6. **Inscribed
angle of a circle is equal to the half of the degree measurement of its
opposite arc.*

*7. **Degree
measurement of a full circle arc is equivalent to 360°.*

For more on central angles, inscribed angles and theorems related to them, visit to the page: Circle Theorems on Central Angles and Inscribed Angles.

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