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Proving a circle theorem

Proving a circle theorem.

  • Give a mathematical reason for each step.
  • Use matching angles and corresponding sides, not the appearance of a diagram.
  • State the result you have proved at the end.

Example 1

Prove that the angle at the centre is twice the angle at the circumference standing on the same arc . Let ACO = α and OCB = β.

Proof of the centre-angle theoremABCO
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Angles around O sum to 360°.Simplify and factor out 2.

Example 2

AB is a diameter and C lies on the circle. Prove that angle ACB is 90°. Let O be the centre.

Proof of the semicircle theoremABCO
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Use the angle sum of triangle ABC.Divide both sides by 2.

Your turn

Question 1

Prove that the angle at the centre is twice the angle at the circumference standing on the same arc . Let ACO = α and OCB = β.

Proof of the centre-angle theoremABCO
Check answer
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Angles around O sum to 360°.Simplify and factor out 2.
Question 2

AB is a diameter and C lies on the circle. Prove that angle ACB is 90°. Let O be the centre.

Proof of the semicircle theoremABCO
Check answer
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Use the angle sum of triangle ABC.Divide both sides by 2.
Question 3

Prove that the angle at the centre is twice the angle at the circumference standing on the same arc . Let ACO = α and OCB = β.

Proof of the centre-angle theoremABCO
Check answer
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Angles around O sum to 360°.Simplify and factor out 2.
Question 4

AB is a diameter and C lies on the circle. Prove that angle ACB is 90°. Let O be the centre.

Proof of the semicircle theoremABCO
Check answer
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Use the angle sum of triangle ABC.Divide both sides by 2.
Question 5

Prove that the angle at the centre is twice the angle at the circumference standing on the same arc . Let ACO = α and OCB = β.

Proof of the centre-angle theoremABCO
Check answer
OA = OC, so triangle AOC is isosceles.OB = OC, so triangle BOC is isosceles.Angles around O sum to 360°.Simplify and factor out 2.
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