Basic Geometry

Standard school geometry: points and angles, triangles and the Pythagorean theorem, polygons and area, similarity, circles, solids, and transformations.

Foundations

• Points, Lines & Planes — Undefined terms, Euclidean postulates, and the betweenness that adds segments up.
• Angles & Angle Pairs — Complementary, supplementary, vertical, and adjacent angles.
• Geometric Proof — Deductive reasoning: justify each step with a postulate or theorem.

Triangles

• Triangle Basics — Classify by sides and angles, and use the 180° angle sum.
• Congruent Triangles — SSS, SAS, ASA, AAS, HL — and why SSA isn't enough.
• Isosceles & Equilateral Triangles — Base angles are equal; an equilateral triangle is all 60°.
• Triangle Inequality — |a + b| ≤ |a| + |b| — and triangle sides obey a + b > c.
• Pythagorean Theorem — a^2 + b^2 = c^2 — the backbone of right-triangle geometry.
• Special Right Triangles — 45-45-90 (1:1: 2) and 30-60-90 (1: 3:2).
• Right-Triangle Trigonometry — SOH-CAH-TOA: find sides and angles with sine, cosine, and tangent.

Parallels, Polygons & Area

• Parallel Lines & Transversals — Corresponding and alternate angles are equal; co-interior angles are supplementary.
• Angles in Polygons — Triangulate to count interior angles; exteriors always total 360^ .
• Quadrilaterals — Parallelograms, trapezoids, and the special parallelograms.
• Midsegments & Medians — A midsegment is half the third side; the centroid splits a median 2:1.
• Regular Polygons & Apothem — Interior angle (n-2)180/n, exterior 360/n, area 12 a P.
• Area & Perimeter — Derive the area formulas — rectangle, parallelogram, triangle, circle — then measure.

Similarity

• Similar Triangles — Two equal angles force the same shape — then side ratios do the rest.
• Proportional Segments — The side-splitter and angle-bisector theorems.
• Areas of Similar Figures — Area scales by the square of the similarity ratio.

Circles

• Central Angles & Arcs — A central angle equals its arc; an inscribed angle is half its arc.
• Inscribed Angle Theorem — Inscribed angle = half the central angle on the same arc.
• Tangents, Chords & Secants — A tangent meets the radius at 90°; intersecting chords multiply equally.
• Circumference, Arc Length & Sectors — C=2π r, arc = θ360 C, sector = θ360 π r^2.
• Equation of a Circle — Center-radius form, reading off center and radius, completing the square.

Solids

• Volume & Surface Area — Derive the solid-volume formulas — stacking, tiling, and Archimedes — then fill them.
• Solid Geometry — Cross-sections, nets, and nested solids — the spatial side of 3D shapes.

Transformations & Constructions

• Locus — The set of all points satisfying a condition.
• Transformations — Translations, reflections, rotations, and dilations — in coordinates.
• Constructions — What you can build with compass and straightedge alone.

Not sure where to start? Take the ten-question placement test.