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Circle Theorem Proofs

Circle Theorem Proofs

The circle theorems can be proved using knowledge of basic angle facts and properties of 2D shapes. If you are asked to prove a circle theorem, adding in any radii and finding all equal angles will help. There are two types of proofs that can be used for the different circle theorems. 

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 x and  y
    • The angle you are trying to prove is 90° is now  x space plus space y
  • STEP 3
    Label the remaining angles in each of the isosceles triangles with algebraic expressions in terms of x and  y
    • Angles at the base of an isosceles triangle are equal
      • Therefore the two remaining angles at the circumference are also  x and  y
    • Angles in a triangle add up to 180°
        • Therefore each angle at the centre will be labelled with the expression  180 space minus space 2 x and  180 space minus space 2 y

4-4-5-circle-theorem-proof-diagram-1

  • STEP 4
    The angles at the centre lie on a diameter, which is a straight line, therefore  180 space minus space 2 x space plus space 180 space minus space 2 y space equals space 180
    • Rearrange this equation to show that  x space plus space y space equals space 90 degree
  • In STEP 2 you already labeled the angle at the circumference as x space plus space y, so this proves that the angle at the circumference equals 90°
    • Give clear reasons throughout your proof, using the key words given in bold

How do I prove that the angle at the centre is twice the angle at the circumference?

  • This circle theorem states that an angle subtended at the centre of a circle is twice the angle subtended at the circumference of a circle from the same arc
  • It does not need any other circle theorems to prove it
  • STEPS 1, 2 and 3 are the same as above

4-4-5-circle-theorem-proof-diagram-2

  • STEP 4
    The three angles formed at the centre lie at a point, therefore they will add to 360°
    • Label the third angle at the centre θ, then you can form the equation theta space plus space 180 space minus space 2 x space plus space 180 space minus space 2 y space equals space 360
    • Rearrange this equation to show that  theta space equals space 2 open parentheses x space plus space y close parentheses
  • In STEP 2 you already labeled the angle at the circumference as x space plus space y, so this proves that the angle at the centre is twice the angle at the circumference
    • Give clear reasons throughout your proof, using the key words given in bold
  • This circle theorem is also a more generic version of the circle theorem the angle in a semicircle is always 90° 
    • You could be asked to prove either without the use of any circle theorems

How do I prove that angles at the circumference from the same arc are equal?

  • This circle theorem states that any angles subtended at the circumference of a circle from the same arc are equal 
  • This theorem is proved using the circle theorem an angle subtended at the centre of a circle is twice the angle subtended at the circumference of a circle
  • Draw the radii from the centre of the circle to the points on the circumference forming the arc which the angles are subtended from
    • This will form an angle at the centre, label this angle 2 x
  • By the circle theorem “The angle and the centre is twice the angle at the circumference”, any angle at the circumference can be labelled x

4-4-5-circle-theorem-proof-diagram-3

  • You do not need to prove the circle theorem you have used in this proof, but you must give clear reasons

How do I prove that opposite angles in a cyclic quadrilateral add up to 180°?

  • This theorem is proved using the circle theorem “An angle subtended at the centre of a circle is twice the angle subtended at the circumference of a circle”
  • STEP 1 Draw the radii from the centre of the circle to any two of the vertices of the cyclic quadrilateral that are opposite each other
    • This will form two angles at the centre, label these angles 2 x and 2 y
    • The angles 2 x and 2 y are at a point, so they add up to 360°
    • Therefore 2 x space plus space 2 y space equals space 360 degree which can be simplified to x space plus space y space equals space 180 degree
  • STEP 2 By the circle theorem “The angle and the centre is twice the angle at the circumference”, the two angles at the circumference can be labelled 1 half open parentheses 2 x close parentheses space equals space x and 1 half open parentheses 2 y close parentheses space equals space y
    • We have already shown that x space plus space y space equals space 180 degree

4-4-5-circle-theorem-proof-diagram-4

  • You do not need to prove the circle theorem you have used in this proof, but you must give clear reasons

How do I prove circle theorems involving chords and tangents?

  • This type of proof can be used to prove the following circle theorems
    • The perpendicular from the centre of a circle bisects a chord
    • The tangent to a circle meets the radius at 90°
    • The alternate segment theorem
  • These proofs can be more tricky and will use other circle theorems within them
  • The questions will often guide you through the proof
  • The circle theorem “The perpendicular from the centre of a circle bisects a chord’ can be proved using congruent triangles
  • If the radius is perpendicular to the chord, two right-angled triangles will be formed  
    • Prove that these two triangle are congruent using the RHS (right angle, hypotenuse, side) rule
      • The chord and the radius are perpendicular, therefore both triangles have a right angle
      • The hypotenuse is the line from the centre to the circumference, therefore both triangles have an equal hypotenuse
      • The line joining the chord to the centre is shared between both triangles, therefore this is a same side in both triangles
      • Therefore, by RHS, the two triangles are congruent

4-4-5-circle-theorem-proof-diagram-5

    • If the two triangles are congruent, then all three sides will be the same and so the perpendicular must bisect the chord
  • The proof for the circle theorem “The tangent to a circle meets the radius at a right angle” uses proof by contradiction and involves assuming that they do not meet at 90° and proving that this is not possible
  • The proof for the alternate segment theorem uses the circle theorems ‘the angle in a semicircle is always 90°’ and ‘the tangent to a circle meets the radius at 90°’

Exam Tip

  • If you are unsure of how to start a proof question, begin by drawing in the radii from the centre to any significant point on the circumference and look for isosceles triangles
  • The question may tell you not to use any circle theorems in your proof, in this case you will most likely be looking for isosceles triangles

Worked example

In the diagram below, AB and C are points on the circumference of a circle, centre O. D C E is a tangent to the circle.

Prove that angle B C E and angle B A C are equal.

0sbebAEi_4-5-5circle-theorem-proof-we-question

Begin by joining the point O to the point C and continue the line through to the circumference so that a diameter is drawn on the diagram.

Label this new point on the circumference F.

Join the point F to the point B on the circumference.

picture14-5-5circle-theorem-proof-we-solution-diagram-1

Angle CBF = 90° The angle in a semicircle is always 90°

The line OC is a radius, so it will meet the tangent DE at 90°.  Let angle B C E space equals space x.

picture14-5-5circle-theorem-proof-we-solution-diagram-2

Angle F C B space equals space 90 space minus space x The radius meets a tangent at 90°

Angles in a triangle add up to 180°, use this to find the angle CFB in terms of x.

table row cell Angle space C F B space end cell equals cell 180 space minus space angle space F C B space minus space angle space C B F end cell row blank equals cell space 180 space minus space open parentheses 90 space minus space x close parentheses space minus space 90 end cell row blank equals cell space 180 space minus space 90 space plus space x space minus space 90 end cell row blank equals cell space x end cell end table The angles in a triangle add up to 180°

Angles subtended at the circumference from the same arc are equal. Use this to find an expression for angle BAC in terms of x.

angle space B A C space equals space angle space space C F B space thin space equals space x degree

We have already stated that  B C E space equals space x.

Therefore Angle BCE  =  Angle BAC  

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