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Foundations of Geometry

Lean on-ramp for the course: precise definitions and notation for points, lines, planes, segments, rays, and angles; segment and angle measurement and addition; coordinate midpoint and distance tools; angle pairs; and the core compass-and-straightedge tool constructions of the cross-cutting geo.construct.* family. Topics: Definitions and Notation, Segment Measurement, Angle Measurement, Angle Pairs, Coordinate Tools: Distance and Midpoint, Core Constructions.

Reasoning and Proof

Proof grammar practiced on easy content: conditional statements and their logical equivalents, inductive versus deductive reasoning, postulates and diagram conventions, and proof formats applied to algebraic, segment, and angle arguments. Topics: Conditional Statements and Logic, Proof Foundations and Formats.

Parallel and Perpendicular Lines

Transversal angle relationships and their proofs, criteria for proving lines parallel or perpendicular, and the coordinate side: slope criteria and equations of parallel and perpendicular lines, placed here with their conceptual basis ahead of the late Coordinate Geometry unit. Topics: Angles Formed by Transversals, Proving Lines Parallel, Perpendicular Lines, Slope Criteria and Equations of Lines.

Transformations

Rigid motions only: performing and describing translations, reflections, and rotations, the properties they preserve, compositions, symmetry, and congruence defined by rigid motions. Topics: Performing and Describing Rigid Motions, Compositions of Rigid Motions, Symmetry, Congruence via Rigid Motions.

Triangle Congruence

The first major synthesis: triangle angle theorems, congruence criteria (SSS, SAS, ASA, AAS, HL) justified by rigid motions, isosceles and equilateral triangle theorems, and congruence proofs with CPCTC. Topics: Triangle Angle Theorems, Congruence Criteria, Congruence Proofs.

Relationships in Triangles

Perpendicular and angle bisectors as loci, the four classical triangle centers, the midsegment theorem, and triangle inequality reasoning including the hinge theorem. Topics: Bisectors as Loci, Triangle Centers, Midsegments, Triangle Inequalities.

Quadrilaterals and Polygons

Polygon angle sums, parallelogram properties and proofs, special parallelograms, trapezoids and kites, and classification within the quadrilateral hierarchy. Topics: Polygon Angle Sums, Parallelograms, Rectangles, Rhombi, and Squares, Trapezoids and Kites, The Quadrilateral Hierarchy.

Similarity

The central relation changes from congruence to proportionality: dilations and their properties open the unit, then similarity defined via transformations, the AA/SSS/SAS criteria, proportionality theorems, and similar right triangles with geometric mean as the bridge into trigonometry. Topics: Dilations, Similar Figures and Criteria, Proportionality Theorems, Similar Right Triangles and Geometric Mean.

Right Triangles and Trigonometry

Pythagorean theorem and converse, special right triangles, and right-triangle trigonometry built on similarity, with elevation and depression applications. Topics: Pythagorean Theorem, Special Right Triangles, Right-Triangle Trigonometry, Trigonometry Beyond Right Triangles.

Circles

Central and inscribed angle relationships, chords, tangents, secants, segment products, circle proofs, arc length, sector area, and the radian connection, plus the circle constructions of the cross-cutting geo.construct.* family. Topics: Circle Vocabulary and Similarity, Angle and Arc Relationships, Chords, Tangents, and Segment Lengths, Circle Proofs, Arc Length, Sectors, and Radians, Circle Constructions.

Coordinate Geometry

The coordinate-methods unit: circle equations derived from the Pythagorean theorem, parabola focus and directrix work, partitioning directed segments, perimeter and area in the plane, and coordinate proofs that combine these tools with earlier figure properties. Topics: Equations of Circles, Parabolas from Focus and Directrix, Partitioning Segments, Coordinate Computations with Figures, Coordinate Proofs.

Solid Geometry and Measurement

Two-dimensional area including regular polygons and composites, cross sections and solids of revolution, volume formulas with informal derivations (including Cavalieri's principle), similar-solid scaling, and density and design modeling as the course's applications capstone. Topics: Area of Polygons and Composite Figures, Cross Sections and Solids of Revolution, Volume, Modeling with Geometry.

Probability

Conditional probability with sample spaces, events, independence, two-way tables, and the addition rule, plus geometric probability and counting methods as further extensions. Topics: Sample Spaces and Events, Independence and Conditional Probability, Geometric Probability and Counting.