Grades 9–12Geometry & TopologyLesson available

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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Builds on

Rigid motions, congruence and similarity The Pythagorean theorem Proof: direct, contrapositive, contradiction and induction

Required prerequisites, all the way down: 21

You will learn to

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.

Definition

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.

Theorem

The angles of a triangle sum to 180°; given the other axioms together with Archimedes' axiom, this is equivalent to the parallel postulate (Dehn's semi-Euclidean planes show Archimedes' axiom is needed).
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.

Common misconceptions

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.

Leads to

Transformation geometry and the Erlangen programme Surfaces, polyhedra and the Euler characteristic Non-Euclidean geometry: spherical and hyperbolic

Optional discoveries

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.
Projective geometry Adding a point at infinity where each family of parallel lines meets gives the projective plane, in which any two distinct lines meet in exactly one point.