Inscribed angles

key notes :

1. Definition of Inscribed Angle:

• An inscribed angle is an angle formed by two chords in a circle that have a common endpoint. This common endpoint is the vertex of the angle, and the sides of the angle are the chords of the circle.

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2. Relationship with the Arc:

• The inscribed angle intercepts a part of the circle’s circumference called the arc.
• The measure of an inscribed angle is half the measure of the intercepted arc.

Formula:

Example: If the intercepted arc is 60°, the inscribed angle will be 30°.

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3. Inscribed Angles Subtended by the Same Arc:

• Angles subtended by the same arc on the circle are equal.
• This means that any two inscribed angles that intercept the same arc will have the same measure.

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4. Inscribed Angle in a Semicircle:

• An inscribed angle that subtends a diameter of the circle is always a right angle (90°).
• This is known as Thales’ Theorem.

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5. Cyclic Quadrilaterals:

• A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle.
• The opposite angles of a cyclic quadrilateral are supplementary (i.e., their sum is 180°).

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6. Application of Inscribed Angles:

• To find the angle between two chords.
• To calculate unknown angles in cyclic quadrilaterals.
• To solve problems involving tangents and secants.

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7. Important Theorems:

• Angle at the Center vs. Angle at the Circumference:
  • The angle at the center of a circle is twice the angle at the circumference subtended by the same arc.

• Angle between a Chord and a Tangent:
  • The angle between a tangent and a chord through the point of contact is equal to the inscribed angle subtended by the chord on the opposite side of the tangent.

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8. Examples:

• If an inscribed angle intercepts an arc of 120°, the angle will measure 60°.
• In a circle with a diameter as a chord, the inscribed angle will always be a right angle (90°).

Learn with an example

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