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

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

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

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

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

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

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 °

Circle Theorems 1.Draw Triangle ABC with B in 3 different positions on the circumference. 2.Measure ABC for each of the 3 triangles. AB 1 C = AB 2 C = AB 3 C = A C 3.Complete the theorem : The angle in a semicircle is

Circle Theorems 32 ° x y 2x2x 3x3x y 72 ° Find the unknown angles.

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’

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. A B x O

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 The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment tangent chord Alternate segment

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

Circle Theorems Exam Question 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 ° (a) Explain why angle OTQ is 90 ° (b) Find the size of the angles (i) TOQ (ii) OPT (c) The angle RTQ is 57 ° Find the size of the angle RUT P Q O T R U S 26 ° [1] [2]

Circle Theorems Additional Notes

Circle Theorems  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 Can Revise Do Further       Learning Outcomes: At the end of the topic I will be able to