Triangle Theorems and Deductive Properties

Understand the logical architecture of geometry and construct rigorous direct proofs, indirect proofs, and counterexamples.

Prove angle relationships involving intersecting and parallel lines and master classic Euclidean compass-and-straightedge constructions.

Prove fundamental triangle theorems including angle sums, inequalities, medians, centroids, and mid-segments.

Establish triangle congruence criteria (SAS, ASA, SSS, AAS, HL) and construct rigorous geometric proofs of triangle properties.

Prove classification properties of special quadrilaterals, derive polygon angle sum formulas, and analyze tessellation conditions.

Prove and apply triangle similarity criteria, Thales' Theorem, area scaling laws, and explore the Golden Ratio.

Prove circle theorems involving chords, central and inscribed angles, angles in the same segment, and cyclic quadrilaterals.

Prove tangent theorems, alternate segment theorem, power of a point, and execute circle tangent constructions.

Apply rigid motions, homothety/dilations, analyze rotational and reflectional symmetries, and calculate coordinate mappings.

Calculate coordinate distances, midpoints, slopes, navigate using three-figure bearings, and interpret 3D orthographic views and nets.