Chapter 3 of seven · Geometry & Algebra
Form
Congruence, quadrilaterals, and the identities as pictures. Meets at (a+b)² as a square.
- What congruence meansThe first lesson of Phase 3 asks a question rather than proving a claim — how much information does it take to pin a shape down?
- SSS, SAS, ASA, agreedThe last lesson asked how much information fixes a shape. This one names the three answers.
- A triangle with two equal sides has two equal anglesThe first payoff of SAS — proving a fact about angles by building a matching pair of triangles, not by measuring.
- The converseTwo equal base angles force two equal sides — proved separately from the last lesson, not assumed from it.
- The perpendicular bisector is the set of points equidistant from two pointsNaming, explicitly, the line the last two lessons already leaned on without drawing it.
- The angle bisector is the set of points equidistant from two linesThe last lesson of Phase 4 — the same locus idea as the one before it, aimed at lines instead of points.
- Parallelograms: opposite sides and angles equalThe first quadrilateral theorem, built from one diagonal and the congruence tools this course has already earned.
- The diagonals of a parallelogram bisect each otherThe second parallelogram theorem, from the same figure as the last lesson and the same ASA move.
- Rectangle, rhombus, square, kite, trapezium as special casesThe parallelogram family tree, and two quadrilaterals that sit outside it entirely.
- The classification, built by the student rather than handed overThe whole quadrilateral family in one figure, with nothing pre-built to already be any of it.