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Circle Theorems

Circle Theorems. Learning Outcomes Revise properties of isosceles triangles, vertically opposite, corresponding and alternate angles Understand the terminology used – angle subtended by an arc or chord Use an investigative approach to find angles in a circle, to include: Angle in a semicircle

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Circle Theorems

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  1. Circle Theorems • Learning Outcomes • Revise properties of isosceles triangles, vertically opposite, corresponding and alternate angles • Understand the terminology used – angle subtended by an arc or chord • Use an investigative approach to find angles in a circle, to include: • Angle in a semicircle • Angle at centre and circumference • Angles in the same segment • Cyclic quadrilaterals • Angle between tangent and radius and tangent kite • Be able to prove and use the alternate segment theorem

  2. Circle Theorems Circle Theorem 1 The angle at the centre of a circle is double the size of the angle at the edge D O A B Angle AOB = 2 x ADB For angles subtended by the same arc, the angle at the centre is twice the angle at the circumference

  3. Circle Theorems Circle Theorem 2 Angles in the same segment are equal C D A B Angle ACB = Angle ADB For angles subtended by the same arc are equal

  4. F D Circle Theorems E 152° O C Circle Theorems Example: Find angle CDE and CFE.

  5. D Circle Theorems O 40° A B Circle Theorems Example: Find giving reasons i) ABO ii) AOB iii) ADB

  6. Circle Theorems D 60° C A B Circle Theorems Example: Find giving reasons i) BAC ii) ABD 38°

  7. Circle Theorems Circle Theorem 3 Opposite angles in a cyclic quadrilateral add up to 180° Angle D + Angle B = 180° Angle A + Angle C = 180° A cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle. The opposite angles add up to 180°

  8. Circle Theorems Circle Theorems • Draw Triangle ABC with B in 3 different positions on the circumference. A • Measure ABC for each of the 3 triangles. • AB1C = • AB2C = • AB3C = • Complete the theorem : C The angle in a semicircle is

  9. y 2x 72° Circle Theorems y 3x 32° x Circle Theorems Find the unknown angles.

  10. Circle Theorems Circle Theorem 4 The angle between the tangent and the radius is 90° The angle between a radius (or diameter) and a tangent is 90 This circle theorem gives rise to one ‘Tangent Kite’

  11. A Circle Theorems x O B Circle Theorems ‘Tangent Kite’ When 2 tangents are drawn from the point x a kite results. The tangents are of equal length BX = AX Given OA = OB (radius) OX is common the, the 2 triangles OAX and OBX are congruent.

  12. Circle Theorems Circle Theorem 5 Alternate Segment Theorem Look out for a triangle with one of its vertices resting on the point of contact of the tangent Alternate segment chord tangent The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment

  13. Circle Theorems Circle Theorem 5 Find all the missing angles in the diagram below, also giving reasons. i) BOA = A C x O 40° B ii) ACB = iii) ABX = iii) BAO =

  14. P T Circle Theorems U Q S O 26° R Exam Question (a) Explain why angle OTQ is 90 ° [1] (b) Find the size of the angles (i) TOQ (ii) OPT [1] [1] (c) The angle RTQ is 57° Find the size of the angle RUT In the diagram above, O is the centre of the circle and PTQ is a tangent to the circle at T. The angle POQ = 90° and the angle SRT = 26° [2]

  15. Circle Theorems Additional Notes

  16. Circle Theorems Learning Outcomes: At the end of the topic I will be able to Can Revise Do Further • Revise properties of isosceles triangles, vertically opposite, corresponding and alternate angles • Understand the terminology used – angle subtended by an arc or chord • Use an investigative approach to find angles in a circle, to include: • Angle in a semicircle • Angle at centre and circumference • Angles in the same segment • Cyclic quadrilaterals • Angle between tangent and radius and tangent kite • Be able to prove and use the alternate segment theorem      

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