An inscribed angle is an angle formed by two chords that meet at a common point on the circle, called the vertex. The angle intercepts an arc, and its measure is equal to half the measure of that arc.
Properties of Inscribed Angles
- Same Arc Property: Inscribed angles that intercept the same arc (or chord) are equal.
- Angle in a Semicircle: An inscribed angle that subtends a diameter is always a right angle (90°).
- Cyclic Quadrilateral Property: In a quadrilateral inscribed in a circle, opposite angles add up to 180° (they are supplementary).
Inscribed Angle Theorem
The Inscribed Angle Theorem states that the measure of an inscribed angle is equal to half the measure of the central angle that subtends the same arc.
- Inscribed Angle: An angle formed by two chords with its vertex on the circle.
- Central Angle: An angle formed by two radii with its vertex at the center of the circle.
- Intercepted Arc: The arc between the endpoints of the angle.
Mathematical Relation
Central Angle = 2 × Inscribed Angle,
Inscribed Angle = (1/2) × Central Angle
Example: If the central angle subtending an arc is 80° then the inscribed angle subtending the same arc is 40°.
Proof
To prove the theorem, we consider three possible positions of the diameter with respect to the inscribed angle.
Case 1: Inscribed Angle Between a Chord and Diameter
To Prove: ∠AOB = 2θ
Proof:
- In △OBD, OB = OD (radii) ⇒ triangle is isosceles
- So, ∠ODB = ∠DBO = θ
- AD is a diameter ⇒ straight line ⇒ ∠BOD = 180° − ∠AOB
Using angle sum of triangle:
θ + θ + (180 − x) = 180⇒ 2θ + 180 − x = 180
⇒ x = 2θTherefore, ∠AOB = 2θ
Case 2: Diameter Between the Rays of the Angle.
To Prove: ∠ACB = 2θ
Proof:
- Split the angle: θ = θ₁ + θ₂
- Corresponding central angles: a = a₁ + a₂
From Case 1:
a₁ = 2θ₁ and a₂ = 2θ₂Adding:
a = a₁ + a₂ = 2θ₁ + 2θ₂ = 2(θ₁ + θ₂)⇒ a = 2θ
Hence, ∠ACB = 2θ
Case 3: Diameter Outside the Rays of the Angle
To Prove: a = 2θ
Proof:
Let θ₁ be the angle at point D and α₁ be its corresponding arc.
From Case 1: α₁ = 2θ₁From the figure, total angle at D = θ₁ + θ
Corresponding total arc = α₁ + αUsing angle–arc relation: 2(θ₁ + θ) = α₁ + α
Substitute α₁ = 2θ₁:
2θ₁ + 2θ = 2θ₁ + αCancel 2θ₁ from both sides: α = 2θ
Hence proved.
Applications of Inscribed Angles
The Inscribed angles are widely used in various mathematical problems and proofs including:
- Proving the properties of the cyclic quadrilaterals.
- Solving the problems related to the circle's geometry such as finding unknown angles and lengths.
- Applications in trigonometry where the angles help establish relationships between the different trigonometric functions.
Related Articles
Solved Examples
Example 1: Given a circle with the central angle ∠AOB = 100° find the measure of the inscribed angle ∠ACB that subtends the same arc AB.
Solution:
Using the Inscribed Angle Theorem:
∠ACB = 1/2 × ∠AOB
∠ACB = 1/2 × 100° = 50°
Example 2: In a circle, points A, B, C and D lie on the circumference forming a cyclic quadrilateral ABCD. If ∠ABC = 70° and ∠BCD = 110°, find the measure of ∠BAD.
Solution:
In a cyclic quadrilateral, opposite angles are supplementary.
So,
∠BAD + ∠BCD = 180°Given,
∠BCD = 110°Therefore,
∠BAD = 180° − 110°
∠BAD = 70°
Example 3: In a circle with center O the diameter AB subtends an inscribed angle ∠ACB. What is the measure of the ∠ACB?
Solution:
Since AB is the diameter ∠ACB is a right angle.
∠ACB = 90°
Example 4: Find the measure of an inscribed angle subtended by the arc measuring 120°.
Solution:
Using the Inscribed Angle Theorem:
∠Inscribed = 1/2 × 120° = 60°
Example 5: If two inscribed angles subtend the same arc and one angle measures 45° what is the measure of the other angle?
Solution:
Since the two inscribed angles subtend the same arc they are equal.
∠1 = ∠2 = 45°
Practice Questions
Question 1. In a circle with center O the diameter AB subtends an inscribed angle ∠ACB. If AB measures 10 cm. what is the measure of ∠ACB?
Question 2. Find the measure of an inscribed angle that subtends an arc of 150°.
Question 3. If two inscribed angles subtend the same arc and one angle measures 35° what is the measure of the other angle?
Question 4. In a circle, the inscribed angle ∠DEF subtends an arc measuring 80°. What is the measure of ∠DEF?
Question 5. In a cyclic quadrilateral ABCD, if ∠ABC = 60° and ∠BCD = 130°, find the measure of ∠BAD.
Question 6. In a circle, the inscribed angle ∠EFD subtends an arc of 120°. Find the measure of ∠EFD.
Question 7. Calculate the measure of the inscribed angle subtended by an arc that measures 200°.
Question 8. A circle has an inscribed angle ∠XYZ that measures 45°. Find the measure of the central angle subtended by the same arc.
Question 9. In a cyclic quadrilateral ABCD, if ∠BCD = 90° and ∠BAD = 80°, find the measure of ∠ABC.
Question 10. Find the measure of an inscribed angle that subtends an arc of 270°.
Answer Key
1. 90° 2. 75° 3. 35° 4. 40° 5. 50°
6. 60° 7. 100° 8. 90° 9. 100° 10. 135°