Geometry

The undefined terms and postulates, the midpoint formula, and the reasoning (conjecture, counterexample, conditionals and converses) that every later proof is written in.

Two-column and paragraph proofs, first used on the angle pairs and parallel-line facts students already know, ending with a proof of the triangle angle sum.

Rigid motions made precise, the symmetries of figures, and proofs that use a transformation as the argument.

Which three facts pin a triangle down (SSS, SAS, ASA, AAS, HL, and why SSA fails), and proofs that use congruent triangles to prove parts equal.

Perpendicular bisectors, angle bisectors, medians and altitudes and the four centers they meet at, then the inequalities between a triangle's sides and angles, proved indirectly.

Angle sums of polygons, then the properties of parallelograms, rectangles, rhombuses, trapezoids and kites, proved and used to classify a quadrilateral.

AA, SSS and SAS similarity proved and used, the triangle proportionality theorem, and how scale factor acts on length, area and volume.

The Pythagorean theorem proved two ways, the geometric mean in a right triangle, the special right triangles, and sine, cosine and tangent as ratios of similar triangles.

Slope, distance and midpoint used to partition segments, find areas, classify figures and prove theorems with a figure placed on axes.

Chords, arcs, inscribed angles, tangents and the segment-length theorems, with arc length, sector area and the equation of a circle.

Compass and straightedge constructions, each one justified by a congruence or circle theorem already proved.

The regular solids, Cavalieri's principle, surface area of pyramids, cones and spheres, and solids of revolution, with composite solids and density reviewed.