Circle-Theorem-Proofs Edexcel Higher

Circle theorem proofs

How do I prove circle theorems using radii to form isosceles triangles?

• This type of proof can be used to prove the following circle theorems

  • The angle in a semicircle is always 90°

  • The angle at the centre is twice the angle at the circumference

  • Angles in the same segment are equal

  • Opposite angles in a cyclic quadrilateral add up to 180°

How do I prove that the angle in a semicircle is 90°?

• This circle theorem states that an angle subtended at the circumference of a semicircle is always a right angle

  • Although it is a special case of the angle at the centre and circumference circle theorem, it can be proved without using any other circle theorems

• STEP 1
  Draw a radius from the centre of the circle to the angle subtended at the circumference

  • This will form two isosceles triangles

• STEP 2
  Label the two angles formed at the angle subtended at the circumference and .

  • The angle you are trying to prove is 90° is now 

• STEP 3
  Label the remaining angles in each of the isosceles triangles with algebraic expressions in terms of and 

  • Angles at the base of an isosceles triangle are equal

    • Therefore the two remaining angles at the circumference are also  and 

  • Angles in a triangle add up to 180°

    • Therefore each angle at the centre will be labelled with the expression  and 

• STEP 4
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