Geometry
Undergraduate · Math
Syllabus focus
Standard syllabus · Theoretical / proof-based
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Topics typically covered
Standard syllabus
Euclidean foundations
- Axiomatic systems; points, lines, and planes
- Angles, parallel lines, and triangle congruence (SAS, ASA, SSS)
- Similarity and proportional segments
- Right triangles; Pythagorean theorem and special triangles
- Circles: chords, tangents, inscribed angles, and arc measure
Coordinate and solid geometry
- Coordinate geometry: distance, midpoint, and slope
- Equations of lines and circles in the plane
- Area and perimeter formulas; surface area and volume of solids
- Cross-sections and nets of 3D figures
- Basic geometric constructions with compass and straightedge
Transformations and symmetry
- Translations, rotations, reflections, and glide reflections
- Congruence and similarity via transformations
- Symmetry groups of regular figures (informal)
- Tessellations and patterns in the plane
- Introduction to non-Euclidean geometry (overview)
Theoretical / proof-based
Axiomatic geometry and proofs
- Incidence geometry and betweenness axioms (Hilbert-style introduction)
- Triangle congruence theorems with full proofs
- Parallel postulate and its role in Euclidean geometry
- Neutral geometry and hyperbolic geometry (overview)
- Saccheri–Legendre theorem and angle-sum results (introduction)
Advanced Euclidean topics
- Ceva's and Menelaus's theorems (optional)
- Nine-point circle and Euler line (optional capstone)
- Inversion in a circle (introduction)
- Projective geometry: points at infinity and duality (preview)
- Geometric transformations as group actions (introduction)
Notes
Topics reflect common college geometry syllabi at US universities. Some programs target future teachers; others serve as a proof-oriented course for math majors. This track list excludes a separate STEM/applied variant.