# GCSE Circle Theorems - Maths with Miss Welton Circle Theorems L.O. All pupils can label parts of circles Most pupils can use prior knowledge to find missing angles Some pupils can derive circle theorems All pupils can solve problems requiring circle theorems RECAP: Parts of a Circle ! (Minor) ? Arc Sector ? Chord

? (Minor) ? Segment Tangent ? Radius ? Diameter ? Circumference ? Circle Theorems L.O.

All pupils can label parts of circles Most pupils can use prior knowledge to find missing angles Some pupils can derive circle theorems All pupils can solve problems requiring circle theorems What are Circle Theorems Circle Theorems are laws that apply to both angles and lengths when circles are involved. Well deal with them in groups. #1 Non-Circle Theorems These are not circle theorems, but are useful in questions involving circle theorems. 130 ? 50 Angles in a quadrilateral add up to 360.

The radius is of constant length Bro Tip: When you have multiple radii, put a mark on each of them to remind yourself theyre the same length. #2 Circle Theorems Involving Right Angles Bro Tip: Remember the wording in the black boxes, because youre often required to justify in words a particular angle in an exam. ! s radiu ent g n ta Angle in semicircle is 90. Angle between radius

and tangent is 90. Note that the hypotenuse of the triangle MUST be the diameter. #3 Circle Theorems Involving Other Angles ! a a a 2a Angles in same segment are equal.

Angle at centre is twice the angle at the circumference. #3 Circle Theorems Involving Other Angles ! x 180-x x Opposite angles of cyclic quadrilateral add up to 180. #4 Circle Theorems Involving Lengths Theres only one you need to know...

Lengths of the tangents from a point to the circle are equal. Which Circle Theorem? Identify which circle theorems you could use to solve each question. Reveal Angle in semicircle is 90 Angle between tangent and radius is 90 Opposite angles of cyclic quadrilateral add to 180 Angles in same segment are equal O Angle at centre is twice angle at circumference

? 160 100 Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360 Which Circle Theorem? Identify which circle theorems you could use to solve each question. Reveal Angle between tangent and radius is 90

Opposite angles of cyclic quadrilateral add to 180 70 60 Angle in semicircle is 90 70 ? Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal

Two angles in isosceles triangle the same Angles of quadrilateral add to 360 Which Circle Theorem? Identify which circle theorems you could use to solve each question. Reveal Angle in semicircle is 90 Angle between tangent and radius is 90 Opposite angles of cyclic quadrilateral add to 180 Angles in same segment are equal 115 ?

Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360 Which Circle Theorem? Identify which circle theorems you could use to solve each question. Reveal Angle in semicircle is 90 Angle between tangent and radius is 90 Opposite angles of cyclic

quadrilateral add to 180 70 ? Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360 Which Circle Theorem? Identify which circle theorems you could use to solve

each question. Reveal Angle in semicircle is 90 Angle between tangent and radius is 90 Opposite angles of cyclic quadrilateral add to 180 32 ? Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles

triangle the same Angles of quadrilateral add to 360 Which Circle Theorem? Identify which circle theorems you could use to solve each question. Reveal Angle in semicircle is 90 Angle between tangent and radius is 90 Opposite angles of cyclic quadrilateral add to 180 31 ? Angles in same segment are equal

Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360 #5 Alternate Segment Theorem This one is probably the hardest to remember and a particular favourite in the Intermediate/Senior Maths Challenges. ! Click to Start Bromanimation This is called the alternate

segment because its the segment on the other side of the chord. o ch rd tangent The angle between the tangent and a chord... ...is equal to the angle in the alternate segment Check Your Understanding

z =?58 Check Your Understanding Source: IGCSE Jan 2014 (R) Angle ABC = 112 ? Give a reason: Supplementary angles of cyclic quadrilateral add up to 180. ? Angle AOC =

136 ? Angle CAE = 68 ? Give a reason: Give a reason: Angle at centre is double angle at circumference. Alternate Segment Theorem.

? ? Circle Theorems L.O. All pupils can label parts of circles Most pupils can use prior knowledge to find missing angles Some pupils can derive circle theorems All pupils can solve problems requiring circle theorems Exercises Questions. Circle Theorems L.O. All pupils can label parts of circles

Most pupils can use prior knowledge to find missing angles Some pupils can derive circle theorems All pupils can solve problems requiring circle theorems Answers to more difficult questions Source: IGCSE May 2013 39 ? 64? 77 ? Determine angle ADB. Answers to more difficult questions

(Towards the end of your sheet) ?1 116 ?322 42 ?3 Angle at centre is twice angle at circumference Two angles in isosceles triangle the same Alternate Segment Theorem Circle Theorems L.O. All pupils can label parts of circles

Most pupils can use prior knowledge to find missing angles Some pupils can derive circle theorems All pupils can solve problems requiring circle theorems APPENDIX: Proofs B ?a A a ? 180-2a O

2a ? 90-a ? ? 90-a C Let angle BAO be a. Triangle ABO is isosceles so ABO = a. Remaining angle in triangle must be 180-2a. Thus BOC = 2a. Since triangle BOC is isosceles, angle BOC = OCB = 90 a. Thus angle ABC = ABO + OBC = a + 90 a = 90. APPENDIX: Proofs ! x

b? This combined angle = 180 a b ? (angles in a triangle) a b a? Opposite angles of cyclic quadrilateral add up to 180. Adding opposite

angles: a + b + 180 a b = 180 APPENDIX: Proofs C 1: Angle between tangent and radius is 90, so angle CAD = 90 - ?3 B 2: Angle in semicircle is 90. ?4

?2 ?1 90- A Alternate Segment Theorem D 3: Angles in triangle add up to 180. 4: But any other angle in the same segment will be the same. Circle Theorems L.O. All pupils can label parts of circles Most pupils can use prior knowledge to find

missing angles Some pupils can derive circle theorems All pupils can solve problems requiring circle theorems