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The Lumenary
الصفوف 9–12Geometry & Topologyمُخطَّط له

Euclidean geometry and proof

Axiomatic plane geometry: congruence criteria, parallel lines, triangle and circle theorems, and writing geometric proofs, with attention to what each axiom contributes.

math.geom.euclidean-geometry

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يعتمد على

Rigid motions, congruence and similarityThe Pythagorean theoremProof: direct, contrapositive, contradiction and induction

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 21

ستتعلّم

  • Use the congruence criteria SAS, ASA and SSS to prove properties of triangles and quadrilaterals.
  • Prove, using the parallel postulate, that the angles of a triangle sum to 180°, and prove circle theorems such as the inscribed angle theorem.
  • Explain the role of the parallel postulate and how its independence led to non-Euclidean geometries.

تعريف

  • Parallel postulate (Playfair's form): through a point not on a given line there is exactly one line parallel to the given line.
  • Two triangles are congruent if their corresponding sides and corresponding angles are equal.
  • An inscribed angle has its vertex on a circle and its sides along chords of the circle.

مبرهنة

  • The angles of a triangle sum to 180°; given the other axioms, this is equivalent to the parallel postulate.
  • Inscribed angle theorem: an inscribed angle is half the central angle subtending the same arc.
  • The perpendicular bisectors of the sides of a triangle meet in a single point, the circumcentre.

مفاهيم خاطئة شائعة

  • Arguing from the appearance of a diagram instead of from stated facts.
  • Treating SSA as a congruence criterion.
  • Believing the parallel postulate has been proved from Euclid's other axioms; it is independent of them.

يقود إلى

Transformation geometry and the Erlangen programme

اكتشافات اختيارية

  • First-order logic The parallel postulate is independent of the other axioms: hyperbolic geometry satisfies them while violating it. Tarski showed that elementary Euclidean geometry, formalised in first-order logic, is complete and decidable.
  • Differential geometry of curves and surfaces On a sphere of radius r the angles of a triangle sum to more than 180°, and the excess (in radians) equals the area divided by r² (Girard's theorem), a first case of Gauss–Bonnet.

مسارات التعلّم

قراءات إضافية

  • Euclid, Elements
  • Robin Hartshorne, Geometry: Euclid and Beyond (2000)
  • A. P. Kiselev, adapted by Alexander Givental, Kiselev's Geometry, Book I: Planimetry