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Proof of Circle Theorems

Arrange the stages of the proofs for the standard circle theorems in the correct order.

Circle Theorems Help Video More on Circles More on Angles

Drag the statements proving the theorem into the correct order.

Circle Theorem
  • Similarly ∠AOC = 180° – 2 x ∠OCA
  • OB = OC (radii of circle)
  • ∠BOA = 2∠BCA Q.E.D.
  • Construct radius OC
  • ∠COB = 180° – 2 x ∠BCO (Angle sum of triangle OBC)
  • To prove: ∠BOA = 2∠BCA
  • ∠BCO = ∠OBC (equal angles in isosceles triangle)
  • ∠BOA = 2(∠BCO + ∠OCA)
  • ∠BOA = 360° – (180° – 2 x ∠BCO + 180° – 2 x ∠OCA)
  • ∴ OBC is an isosceles triangle

Correct Wrong

Circle Theorem
  • ∴ 2 x ∠ABD = 2 x ∠ACD
  • ∠AOD = 2 x ∠ABD (angle at centre twice angle at circumference)
  • ∠AOD = 2 x ∠ACD (angle at centre twice angle at circumference)
  • ∠ABD = ∠ACD Q.E.D.
  • Construct radii from A and D
  • To prove: ∠ABD = ∠ACD

Correct Wrong

Circle Theorem
  • Similarly in triangle BCO ∠OCB = ∠OBC
  • To prove: ∠ABC = 90°
  • ∴ ∠OAB = ∠OBA (equal angles in isosceles triangle ABO)
  • ∠OAB + ∠OBA + ∠OCB + ∠OBC = 180° (Angle sum of triangle ABC)
  • OA = OB (radii)
  • ∴ 2(∠OBA + ∠OBC) = 180°
  • ∴ ABO is an isosceles triangle (two equal sides)
  • ∠ABC = 90° Q.E.D.
  • Construct the radius OB
  • ∴ ∠OBA + ∠OBC = 90°

Correct Wrong

Circle Theorem
  • The obtuse and reflex angles at O add up to 360° (angles at a point)
  • Similarly the obtuse angle AOC = 2 x ∠CDA
  • To prove ∠ABC + ∠CDA = 180°
  • ∴ 2 x ∠ABC + 2 x ∠CDA = 360°
  • Reflex ∠AOC = 2 x ∠ABC (angle at centre twice angle at circumference)
  • ∠ABC + ∠CDA = 180° Q.E.D.
  • Construct the radii OA and OC

Correct Wrong

Circle Theorem
  • 2 x ∠CAB = 2 x ∠CBD (from [1] above)
  • ∠OBC + ∠CBD = 90° (angle between radius and tangent) [2]
  • ∠CAB = ∠CBD Q.E.D.
  • 2 x ∠OBC + ∠COB = 180° (angle sum of triangle) [3]
  • Obtuse ∠COB = 2 x ∠CAB (angle at centre twice angle at circumference) [1]
  • ∠COB = 2 x ∠CBD
  • To prove ∠CAB = ∠CBD
  • ∠OBC = ∠OCB (equal angles in isosceles triangle OBC)
  • Construct the radii OB and OC
  • 2 x ∠OBC + ∠COB = 2(∠OBC + ∠CBD) (from [2] and [3] above)

Correct Wrong


This is Proof of Circle Theorems level 0. You can also try:


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Proof of Circle Theorems

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