Proof and circle geometry.
Proof and circle geometry: statements, implication, converse and equivalence, counterexamples, direct proof and contradiction.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Statements, implication, converse and equivalence
Separate an assumption from a conclusion, then decide whether a converse or an equivalence is justified.
- 02
Counterexamples
Use one carefully chosen example to disprove a universal claim, while recognising what examples cannot prove.
- 03
Direct proof and contradiction
Build general arguments from definitions, and use contradiction when an assumption forces an impossibility.
- 04
Rational/irrational numbers and simple number proofs
Use precise definitions of rational and irrational numbers to prove general statements and identify false ones.
- 05
Interval notation
Describe real solution sets precisely using intervals, unions and intersections.
- 06
Angle properties of circles
Connect central, inscribed and semicircle angles by identifying the exact arc an angle subtends.
- 07
Cyclic quadrilaterals
Recognise cyclic quadrilaterals, calculate their angles and use the converse theorem in a proof.
- 08
Chords and tangents
Use radii, perpendicular bisectors and tangent relationships to turn a circle sketch into a proof or calculation.
- 09
Intersecting-chord/secant/tangent length relationships
Choose the correct segment product for intersecting chords, secants and tangents.
- 10
Congruence and similarity in geometric proofs
Build geometric proofs by establishing congruence or similarity before claiming equal or proportional lengths.
Topic review
Revisit the key ideas, then choose a practice paper.