9.º a 12.º gradoGeometry & TopologyPlanificado
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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Se basa en
Rigid motions, congruence and similarityThe Pythagorean theoremProof: direct, contrapositive, contradiction and induction
Requisitos previos obligatorios, hasta el fondo: 21
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- 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.
Conduce a
Descubrimientos opcionales
- 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.
Itinerarios de aprendizaje
Lecturas complementarias
- Euclid, Elements
- Robin Hartshorne, Geometry: Euclid and Beyond (2000)
- A. P. Kiselev, adapted by Alexander Givental, Kiselev's Geometry, Book I: Planimetry