What is the Inscribed Angle Theorem formula proof and solved examples
Inscribed Angle Theorem is a must-know rule in geometry, especially for CBSE and competitive exams. It connects angles formed by chords and arcs, helping students solve tricky circle questions efficiently. Mastering this theorem boosts confidence in board problems and real-world scenarios involving circles.
Formula Used in Inscribed Angle Theorem
The standard formula is: \( \text{Inscribed Angle} = \frac{1}{2} \times \text{Central Angle subtending the same arc} \)
Here’s a helpful table to understand Inscribed Angle Theorem more clearly:
Inscribed Angle Theorem Table
| Term | Definition | Circle Example |
|---|---|---|
| Inscribed Angle | Angle with vertex on the circle, sides are chords | Angle at point C on circle |
| Central Angle | Angle at circle’s center, sides are radii | Angle at center O |
| Intercepted Arc | Arc between endpoints of angle’s sides | Arc AB |
This table shows how the pattern of Inscribed Angle Theorem appears regularly in real exam questions and diagrams.
Worked Example – Solving a Problem
Let's solve a classic circle question using the inscribed angle theorem. Suppose, in a circle, the central angle ∠AOB is 120°. Find the inscribed angle ∠ACB that subtends the same arc AB.
1. Write the relationship:2. Substitute:
3. Final Answer:
You can practice similar problems using more theorems from our Seven Circle Theorems page for a deeper understanding of circle geometry.
Practice Problems
- If the central angle subtending arc PQ is 80°, what is the measure of the inscribed angle subtending the same arc?
- In a circle, an inscribed angle is 45°. What is the central angle subtending the same arc?
- True or False: All inscribed angles subtending the same arc are equal.
- If the inscribed angle in a semicircle is ___°, fill in the blank.
Common Mistakes to Avoid
- Confusing Inscribed Angle Theorem with the central angle theorem—they are closely related, but not the same. Read about the differences on the Circle Theorem page.
- Forgetting that all inscribed angles subtending the same arc are equal, regardless of their position on the circumference.
- Mixing up arc or chord names—always double-check which angle is being subtended by which arc or chord.
Real-World Applications
The concept of Inscribed Angle Theorem appears in fields like engineering design, clockmaking, and construction—anywhere angles in circles are involved. For more about practical circle uses and properties, visit Properties of Circle. Vedantu makes it easy to connect maths theory with everyday logic.
We explored the idea of Inscribed Angle Theorem, its formula, solved examples, and how it applies both in classroom and the real world. Practicing with Vedantu’s maths resources helps you master such key theorems for exams and beyond.
FAQs on Inscribed Angle Theorem in Circle Geometry Explained
1. What is the Inscribed Angle Theorem?
The Inscribed Angle Theorem states that an inscribed angle in a circle is equal to half the measure of its intercepted arc. An inscribed angle is formed by two chords that meet at a point on the circle. If the intercepted arc measures 100°, then the inscribed angle measures 50°. This theorem is a fundamental result in circle geometry and is widely used in solving angle and arc problems.
2. What is the formula for the Inscribed Angle Theorem?
The formula for the Inscribed Angle Theorem is m∠A = ½ × m(arc).
- m∠A represents the measure of the inscribed angle.
- m(arc) represents the measure of the intercepted arc.
3. How do you find an inscribed angle in a circle?
To find an inscribed angle, calculate half of the intercepted arc.
- Step 1: Identify the arc intercepted by the angle.
- Step 2: Measure or determine the arc’s degree value.
- Step 3: Multiply the arc measure by ½.
4. Why is an inscribed angle half the arc measure?
An inscribed angle is half the arc measure because it subtends the same arc as a central angle, and a central angle equals the full arc measure. Since the central angle is twice the inscribed angle, the inscribed angle must be ½ of the intercepted arc. This relationship can be proven using triangle properties and circle symmetry.
5. What is the difference between an inscribed angle and a central angle?
The main difference is that a central angle equals the arc measure, while an inscribed angle equals half the arc measure.
- A central angle has its vertex at the center of the circle.
- An inscribed angle has its vertex on the circle.
- Central angle formula: m∠ = arc.
- Inscribed angle formula: m∠ = ½ × arc.
6. Can you give an example of the Inscribed Angle Theorem?
Yes, if an arc measures 150°, the inscribed angle that intercepts it measures 75°. Using the formula m∠ = ½ × arc:
- Given arc = 150°
- Inscribed angle = ½ × 150°
- = 75°
7. What happens when an inscribed angle intercepts a semicircle?
When an inscribed angle intercepts a semicircle, it forms a right angle (90°). A semicircle measures 180°, so applying the theorem gives ½ × 180° = 90°. This result is sometimes called Thales’ Theorem and is a special case of the Inscribed Angle Theorem.
8. Are inscribed angles that intercept the same arc equal?
Yes, inscribed angles that intercept the same arc are congruent (equal in measure). Since each angle equals half of the same intercepted arc, their measures must be equal. This property is often used to prove angles equal in cyclic quadrilaterals and circle geometry proofs.
9. How is the Inscribed Angle Theorem used in solving circle problems?
The Inscribed Angle Theorem is used to find missing angles or arc measures in circle geometry.
- Find a missing inscribed angle using ½ × arc.
- Find a missing arc using 2 × inscribed angle.
- Prove angles are equal in cyclic figures.
10. What are common mistakes when using the Inscribed Angle Theorem?
A common mistake is forgetting that the inscribed angle equals half the arc measure, not the full arc.
- Confusing central and inscribed angles.
- Using the wrong intercepted arc.
- Forgetting that the vertex must lie on the circle.
