Angles in a Circle

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A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (centre) on the plane. In other words, a circle is the locus of a point that moves in a plane so that its distance from a fixed point in the same plane always remains constant. An angle formed between the radii, chords, or tangents of a circle is known as the angle of a circle.

Let’s understand properties related to the angle in a circle and theorems based on the angle properties.

Let’s first start with an angle subtended by a chord at a point in a circle.

Consider the following figure.

In the above figure, $\text{PQ}$ is a chord in a circle with centre $\text{O}$. There are three angles in the above figure.

$\angle \text{PRQ}$
$\angle \text{POQ}$
$\angle \text{PSQ}$

All these angles are subtended by the chord $\text{PQ}$.

$\angle \text{PRQ}$ and $\angle \text{PSQ}$ are subtended by the chord $\text{PQ}$ at the two points $\text{P}$ and $\text{S}$ on the circle. These angles are known as angles subtended by the chord at a point on a circle.

$\angle \text{POQ}$ is subtended by the chord $\text{PQ}$ at the centre $\text{O}$ of the circle. This angle is known as an angle subtended by the chord at the centre of the circle.

Let’s further understand the properties of these two types of angles, i.e.,

Angle subtended by a chord at a point on a circle
Angle subtended by a chord at the centre of a circle

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Angles in the Same Segment of a Circle are Equal

One theorem in angles in a circle is related to the angles in the same segment of a circle are equal is equal chords of a circle subtend equal angles at the centre.

Theorem: Equal chords of a circle subtend equal angles at the centre

In the above figure, you are given two equal chords $\text{AB}$ and $\text{CD}$ of a circle with centre $\text{O}$ and you want to prove that $\angle \text{AOB} = \angle \text{COD}$.

In triangles $\text{AOB}$ and $\text{COD}$,

$\text{OA} = \text{OC}$ (Radii of a circle)

$\text{OB} = \text{OD}$ (Radii of a circle)

$\text{AB} = \text{CD}$ (Given)

Therefore, $\triangle \text{AOB} \cong \triangle \text{COD}$ (SSS rule)

Thus, $\angle \text{AOB} = \angle \text{COD}$ (Corresponding parts of congruent triangles)

Now, let’s understand the converse of the above theorem, i.e., if the angles subtended by the chords of a circle at the centre

are equal, then the chords are equal.

Again, let’s refer to the same figure.

Here, it is given that $\angle \text{AOB} = \angle \text{COD}$, and you want to prove that $\text{AB} = \text{CD}$.

Consider $\triangle \text{AOB}$ and $\triangle \text{COD}$

$\text{OA} = \text{OC}$ (Radii of a circle)

$\text{OB} = \text{OD}$ (Radii of a circle)

$\angle \text{AOB} = \angle \text{COD}$ (Given)

Therefore, $\triangle \text{AOB} \cong \triangle \text{COD}$

Thus, $\text{AB} = \text{CD}$ (Corresponding parts of congruent triangles)

Angle Subtended By a Chord at the Centre

Theorem: The angle subtended by an arc (a chord) at the centre is double the angle subtended by it at any point on the remaining part of the circle.

The above figure shows three cases where an arc (or a chord) $\text{PQ}$ subtendes an angle at the centre $\angle \text{POQ}$ and at any point $\text{A}$ on the circle.

The three cases shown in the figure are

Minor arc $\text{PQ}$
Semicircle $\text{PQ}$
Major arc $\text{PQ}$

Let’s join $\text{AO}$ and extend it to the point $\text{B}$. (in all the three cases).

In all the cases, $\angle \text{BOQ} = \angle \text{OAQ} + \angle \text{AQO}$ (An exterior angle of a triangle is equal to the sum of the two interior opposite angles)

Also in $\triangle \text{OAQ}$

$\text{OA} = \text{OQ}$ (Radii of a circle)

$\angle \text{OAQ} = \angle \text{OQA}$ (Using the theorem ‘If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.’)

Therefore, $\angle \text{BOQ} = 2 \angle \text{OAQ}$ ——————- (1)

Similarly, you can prove that $\angle \text{BOP} = 2 \angle \text{OAP}$ —————- (2)

From (1) and (2), you get $\angle \text{BOP} + \angle \text{BOQ} = 2 \left( \angle \text{OAP} + \angle \text{OAQ} \right)$

which is the same as $\angle \text{POQ} = 2 \angle \text{PAQ}$ —————- (3)

For the case 3(Major arc), where $\text{PQ}$ is the major arc, (3) is replaced by reflex angle $\text{POQ} = 2 \angle \text{PAQ}$.

Angles in a Circle

Key Takeaways

Inscribed angles subtended by the same arc are equal.
Central angles subtended by arcs of the same length are equal.
The central angle of a circle is twice any inscribed angle subtended by the same arc.
An angle inscribed in a semicircle is $90^{\circ}$.
An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
The opposite angles of a cyclic quadrilateral are supplementary
The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa.
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
When two segments are drawn tangent to a circle from the same point outside the circle, the segments are equal in length.

Famous Mathematicians

Practice Problems

What is meant by an angle inscribed in a circle?
What is meant by a central angle of a chord of a circle?
What is the relation between an angle subtended by a chord of a circle and its central angle?

FAQs

What is meant by an angle in a semicircle?

An angle subtended by the diameter of a circle at any point of the circle is called the angle in a semicircle.

What is the measure of an angle in a semicircle?

The measure of an angle in a semicircle or an angle subtended by the diameter of a circle at any remaining part of it is $90^{\circ}$.

Are the angles subtended by two equal chords of a circle at the centre are equal?

Yes, the angles subtended by two equal chords of a circle at the centre are equal.

Conclusion

The angle subtended by a chord at any part of the circle is called the angle inscribed by the chord at a point and the angle subtended by the chord at the centre of the circle is called the central angle. The central angle is always double the measure of an angle subtended by the chord at any remaining part of the circle.

Angles in a Circle – Meaning, Properties & Examples