Circles and Angles

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The inscribed angle is half the central angle subtending the same arc. Formula: Inscribed angle = (1/2) × Central angle Example: If central angle = 120°, then inscribed angle = 60° This relationshi…

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You should use a 60° angle. Step-by-step reasoning: 1. 1/3 of circle = central angle of 120° 2. Using the inscribed angle theorem: inscribed angle = (1/2) × central angle 3. Required inscribed angle …

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An angle inscribed in a semicircle is always 90° (a right angle). Proof: - Semicircle has central angle = 180° - Inscribed angle = (1/2) × 180° = 90° This is known as Thales' theorem and is very use…

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The angle at a point on the major arc = 40° Solution steps: 1. Given: Central angle = 80° 2. The point is on the circle but not on the arc (on major arc) 3. Apply inscribed angle theorem: Inscribed a…

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Alternate arcs: Two arcs formed when any two points divide a circle. Key relationships: 1. Angles inscribed in the same arc are equal 2. Sum of angles inscribed in alternate arcs = 180° Example: If …

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Cyclic Quadrilateral: A quadrilateral whose all four vertices lie on a circle. Main Property: Sum of opposite angles = 180° Formally: ∠A + ∠C = 180° and ∠B + ∠D = 180° Converse: If sum of opposite …

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Solution: Given: ∠A = 75°, ∠B = 110° Step 1: Use property of cyclic quadrilateral ∠A + ∠C = 180° (opposite angles) ∠B + ∠D = 180° (opposite angles) Step 2: Calculate ∠C 75° + ∠C = 180° ∠C = 180° - 7…

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Method using 'Angle in semicircle = 90°': Steps: 1. Draw a circle passing through the given point on the line 2. Ensure the circle intersects the line at another point 3. Draw the diameter through th…