How Are Inscribed and Central Angles Related in Circles?
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 Understanding the Inscribed Angle Theorem: Complete Guide for Students
1. What is the Inscribed Angle Theorem?
The Inscribed Angle Theorem states that an inscribed angle in a circle is always half the measure of the central angle subtending the same arc. In simpler terms, if an angle is formed by two chords meeting at a point on the circumference, its measure is half of the corresponding central angle that subtends the same arc. This is a key theorem in class 9 and class 10 geometry.
2. What is the formula for the Inscribed Angle Theorem?
Formula: If ∠ABC is an inscribed angle that subtends arc AC, and ∠AOC is the central angle subtending the same arc, then:
∠ABC = ½ × ∠AOC
Where O is the centre of the circle. This formula helps to quickly calculate unknown angles in circle geometry problems.
3. What is the Inscribed Angle Theorem 90?
If an inscribed angle in a circle subtends a diameter, it must be a right angle (i.e., 90°). This is a special case of the Inscribed Angle Theorem often called Thales' Theorem. In other words, any angle formed in a semicircle is always a right angle.
4. What is the standard form of the Inscribed Angle Theorem for class 10?
In Class 10, the Inscribed Angle Theorem is generally stated as: The measure of an inscribed angle is half the measure of the central angle subtending the same arc. This theorem is used for finding unknown angles and proving properties related to circles.
5. How can you use the Inscribed Angle Theorem to justify an angle?
You can use the Inscribed Angle Theorem to justify that an angle drawn from the circumference to the endpoints of the same arc will always measure half as much as the central angle over that arc. This helps in proving angle relationships, especially in MCQs and geometry proofs.
6. What is the rule for inscribed and central angles in a circle?
The rule is: The inscribed angle is always half the measure of the central angle that subtends the same arc. This fundamental property helps relate all angles subtending the same arc, allowing multiple geometry problems to be solved easily.
7. What is the proof of the Inscribed Angle Theorem?
To prove the Inscribed Angle Theorem:
1. Draw a circle with centre O and inscribed angle ∠ABC.
2. Join OA, OB, and OC.
3. Show that ∠AOC (central angle) = 2 × ∠ABC.
4. Use properties of isosceles triangles (since OA = OB = OC = radius) and linear pairs.
Thus, ∠ABC = ½ × ∠AOC.
8. What are some examples using the Inscribed Angle Theorem?
Example: If a circle has a central angle of 120°, the inscribed angle over the same arc will be 60°, since 120° ÷ 2 = 60°. Conversely, if you know the inscribed angle is 35°, the central angle over the same arc must be 70°.
9. What is the Central Angle Theorem?
The Central Angle Theorem states that the measure of a central angle is equal to the measure of its subtended arc. This is important in differentiating central from inscribed angles.
10. What is the Inscribed Angle Theorem 1?
In some textbooks, Inscribed Angle Theorem 1 refers to the statement: An angle inscribed in a semicircle is a right angle. This is a direct application of the general Inscribed Angle Theorem.
11. Where can I find an Inscribed Angle Theorem Worksheet or Practice Problems?
You can practice Inscribed Angle Theorem questions using worksheets available in your NCERT textbook, class 9/10 geometry modules, or downloadable PDF worksheets from educational sites like Vedantu. These include a variety of practice problems on calculating angles and using this theorem for proofs.
12. What does the Inscribed Angle Theorem mean in simple terms?
In simple terms, the Inscribed Angle Theorem means that if you draw an angle from the edge of a circle, its measure is always half the angle at the centre made by lines from the same ends of the arc.
