Angle Properties of a Circle

Angle properties are fundamental in geometry, particularly in understanding the relationships between angles formed by chords, arcs, and tangents in a circle. This section covers two key properties:

• Angle at the Center: The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.
• Angles in the Same Segment: Angles in the same segment of a circle (i.e., angles subtended by the same chord on the same side) are equal.

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Formulae

The core formulas are described by these relationships:

• If an arc subtends an angle $x$ at the center, it subtends an angle $x/2$ at any point on the major arc.
• If $\angle APB$ and $\angle AQB$ are angles in the same segment, then $\angle APB = \angle AQB$. (A, B are points on the circumference and P, Q are points on the same side of AB)

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Examples

Example-1:

Consider a circle with center O. If arc AB subtends an angle of $100^\circ$ at the center O (i.e., $\angle AOB = 100^\circ$), then the angle subtended by arc AB at a point C on the circumference (on the major arc) will be $\angle ACB = 100^\circ / 2 = 50^\circ$.

Example-2:

In a circle, let’s say we have a chord AB. If $\angle APB = 60^\circ$ and points P and Q lie on the same side of AB on the circumference, then $\angle AQB$ will also be $60^\circ$.

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Theorem with Proof: Angle at Centre is Double Angle at Circumference

Theorem: The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.

Proof:

1. Case 1: When the center of the circle, O, lies on one of the arms of the angle at the circumference (e.g., on AP, where C is the point on circumference).
   1. Join CO. Then OA = OC (radii of the same circle). Therefore, $\triangle OAC$ is isosceles, and $\angle OAC = \angle OCA$.
   2. Let $\angle OCA = x$. Then $\angle OAC = x$. By the exterior angle property, $\angle COB = \angle OAC + \angle OCA = x + x = 2x$.
   3. So, $\angle COB = 2\angle CAB$. The angle at the center is double the angle at the circumference.
2. Case 2: When the center of the circle, O, lies inside the angle at the circumference.
   1. Draw the diameter through A and let the point where it meets the circle be D.
   2. Then, using Case 1: $\angle DOB = 2\angle DAB$ and $\angle DOA = 2\angle DAC$.
   3. Adding, $\angle DOB + \angle DOA = 2\angle DAB + 2\angle DAC$.
   4. Hence, $\angle BOA = 2\angle BAC$. The angle at the center is double the angle at the circumference.
3. Case 3: When the center of the circle, O, lies outside the angle at the circumference.
   1. Draw the diameter through A and let the point where it meets the circle be D.
   2. Then, using Case 1: $\angle DOB = 2\angle DAB$ and $\angle DOA = 2\angle DAC$.
   3. Subtracting, $\angle DOB – \angle DOA = 2\angle DAB – 2\angle DAC$.
   4. Hence, $\angle BOA = 2\angle BAC$. The angle at the center is double the angle at the circumference.

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Common Mistakes by Students

• Incorrect Identification of Angles: Misidentifying the angle at the center or the angle at the circumference. Carefully identify the arc that subtends the angles.
• Assuming Equality Instead of Doubling: Forgetting that the angle at the center is double the angle at the circumference, not equal.
• Applying the Property Incorrectly: Applying the properties to angles that are not subtended by the same arc or chord. Always ensure the arc or chord is the same.

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Real Life Application

Angle properties are fundamental to the design and functionality of many systems, for example:

• Architecture: Understanding angle properties is key to designing circular structures like domes, arches, and circular buildings.
• Navigation: In surveying and navigation, these concepts are used in determining locations using bearings and angles.
• Engineering: Used in the design of gears, where the angles subtended at the center play a crucial role in the ratio of gear rotation.

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Fun Fact

The ancient Greek mathematician Thales of Miletus is credited with the theorem that any angle inscribed in a semicircle is a right angle (a special case of the angles in the same segment property). This was one of the earliest proofs in geometry!

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Recommended YouTube Videos for Deeper Understanding

Q.1 In a circle, $O$ is the center. $A$, $B$, and $C$ are points on the circumference. If $\angle AOB = 100^\circ$, what is the measure of $\angle ACB$?

Check Solution

Ans: B

$\angle ACB = \frac{1}{2} \angle AOB$ (Angle at the center is double the angle at the circumference). Therefore, $\angle ACB = \frac{1}{2} \times 100^\circ = 50^\circ$.

Q.2 In a circle, $P$, $Q$, $R$, and $S$ are points on the circumference such that $PQ$ and $RS$ form an arc. If $\angle PSR = 70^\circ$, what is the measure of $\angle PQR$?

Check Solution

Ans: C

Angles in the same segment are equal, and $\angle PSR$ and $\angle PQR$ subtend opposite arcs so are not in the same segment. The sum of opposite angles in a cyclic quadrilateral is $180^\circ$, so $\angle PQR = 180^\circ – \angle PSR = 180^\circ – 70^\circ = 110^\circ$.

Q.3 In a circle, $O$ is the center. $X$ and $Y$ are points on the circumference, and $Z$ is a point on the circumference such that $XZY$ is a major arc. If $\angle XOY = 120^\circ$, what is the reflex angle $\angle XZY$?

Check Solution

Ans: D

The angle at the major arc is half of the angle at the centre. Thus, reflex $\angle XZY = 360^\circ – \frac{1}{2}\angle XOY$. That is, $\angle XZY = \frac{1}{2} \times 120^\circ = 60^\circ$. Therefore, reflex $\angle XZY = 360^\circ – 60^\circ = 300^\circ$.

Q.4 In a circle, $A$, $B$, $C$, and $D$ are points on the circumference such that $AB$ and $CD$ form the same arc. If $\angle BAC = 35^\circ$, what is the measure of $\angle BDC$?

Check Solution

Ans: A

$\angle BAC$ and $\angle BDC$ are angles in the same segment, so they are equal. Therefore, $\angle BDC = 35^\circ$.

Q.5 In a circle with center $O$, points $L$, $M$, and $N$ lie on the circumference. If $\angle LMN = 65^\circ$, and $LO$ and $NO$ are radii, find the measure of $\angle L O N$ at the centre.

Check Solution

Ans: B

Since $\angle LMN$ is an angle on the circumference, the corresponding angle at the center is double, so the reflex angle $\angle L O N = 2 \times 65^\circ = 130^\circ$. The angle $\angle LON$ is the reflex angle subtracted from 360. In this case, the angle is $130^\circ$.

Next Topic: Cyclic Quadrilaterals (Class 9)