Euclidean and coordinate geometry foundations
Twelve lessons in high-school Euclidean and coordinate geometry: diagrams, constructions, calculations and written arguments. This foundations unit is not the complete geometry and trigonometry curriculum; solid geometry, conics and the laws of sines and cosines follow separately.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
Loading saved mathematics work…
Geometry foundations: calculation and proof assessment
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Separate a drawing from a geometric argument
Lesson details and objectives
Use incidence, betweenness and definitions to justify geometric claims.
- Distinguish definitions, assumptions and proved conclusions.
- State the conditions behind segment and angle addition.
6 practice tasks.
Track angles through a transversal
Lesson details and objectives
Connect vertical and supplementary angles with theorems about parallel lines.
- Use a parallel-line hypothesis at the correct step.
- Distinguish an angle theorem from its converse.
6 practice tasks.
Match triangles with sufficient evidence
Lesson details and objectives
Use correspondence and congruence criteria without relying on scale or orientation.
- Apply SSS, SAS and ASA with explicit vertex correspondence.
- Explain why two sides and a nonincluded angle can be ambiguous.
6 practice tasks.
Locate triangle centers and test possible sides
Lesson details and objectives
Distinguish medians, bisectors and altitudes, and recognize degenerate side data.
- Identify which lines define each center.
- Apply strict triangle inequalities and centroid ratios.
6 practice tasks.
Move a figure without changing its shape
Lesson details and objectives
Track translations, reflections and rotations using coordinates and distance preservation.
- Calculate images under elementary rigid motions.
- Justify congruence through preservation of distances.
6 practice tasks.
Scale lengths, areas and arguments
Lesson details and objectives
Use equal angles and proportional sides to distinguish similarity from congruence.
- Find and apply a consistent similarity ratio.
- Derive area scaling and account for the dilation center.
6 practice tasks.
Connect right angles with squared lengths
Lesson details and objectives
Prove the Pythagorean relation and use its converse without assuming the desired right angle.
- Identify the hypotenuse and compute exact lengths.
- Separate the theorem, its converse and angle classification.
6 practice tasks.
Choose ratios relative to an angle
Lesson details and objectives
Use similarity to explain why sine, cosine and tangent depend on an acute angle.
- Identify opposite, adjacent and hypotenuse relative to a named angle.
- Apply and justify ratios with explicit geometric assumptions.
6 practice tasks.
Decompose a region without counting it twice
Lesson details and objectives
Justify polygon angle sums and area formulas by partitioning into simpler shapes.
- Use a valid triangulation for angle sums.
- Distinguish perpendicular height, side length and signed decomposition.
6 practice tasks.
Read the arc before using the angle
Lesson details and objectives
Relate central and inscribed angles, chords and tangent lengths with explicit circle hypotheses.
- Identify the intercepted arc of an inscribed angle.
- Use equal radii and perpendicular tangents in proofs.
6 practice tasks.
Choose coordinates that expose the geometry
Lesson details and objectives
Translate geometric assumptions into algebra while preserving generality and exceptional cases.
- Use midpoint, distance and vector relations in proofs.
- Avoid dividing by zero when discussing vertical lines.
6 practice tasks.
Construct a point and justify why it works
Lesson details and objectives
Separate a permitted construction procedure from its mathematical correctness.
- Describe compass-and-unmarked-straightedge steps precisely.
- Prove perpendicular and angle bisectors using distance and congruence.
6 practice tasks.