Mathematics · unit version 1

Geometric definitions, motion witnesses and theorem proofs

A school-entry reasoning unit deriving congruence and similarity criteria from explicit Euclidean motion and incidence facts, then proving parallel-line and triangle/quadrilateral results. Written proofs and constructions require review; public completion is not a mastery certification.

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

Check your starting point

Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.

What is the difference between a definition and a theorem?

A proof may use stated assumptions and established results, but not its own desired conclusion.

Compare reasoning and find a refresher

A definition fixes the meaning of a term. A theorem asserts a consequence that needs justification from assumptions. A diagram illustrates a configuration; appearance alone is not an assumption.

Review definitions and geometric reasoning

How do you compute the distance between two coordinate points?

Exact squared distances help check motion witnesses and rule out impossible matches.

Compare reasoning and find a refresher

Subtract the horizontal and vertical coordinates; add their squares. The distance is the nonnegative square root. Equality of squared distances suffices to compare lengths without rounding.

Review coordinate distance

What does a correspondence between triangles specify?

Side ratios and angle equalities must use the same vertex pairing.

Compare reasoning and find a refresher

A↔D, B↔E, C↔F pairs AB with DE, BC with EF and CA with FD. Reversing orientation is allowed, but changing the pairing in the middle of an argument is not.

Review corresponding parts

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Construct witnesses and justify geometric conclusions

The mixed assessment has separate response history and review criteria; it does not change lesson participation records.

Lesson sequence

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Make geometric definitions testable
Lesson details and objectives
Use precise definitions and counterexamples before trusting a picture.
- Distinguish primitive points and lines from defined segments, rays and angles.
- Use distance, incidence and coplanarity conditions in circle and line definitions.
- Refute inaccurate definitions with explicit coordinate counterexamples.
6 practice tasks.
Treat a motion as a function on the plane
Lesson details and objectives
Track every point through translations, rotations and reflections.
- Specify a motion by its defining parameters and map points accordingly.
- Explain preservation of distances and angles for rigid motions.
- Compare rigid motions with a nonuniform stretch and track composition order.
6 practice tasks.
Find the complete symmetry set
Lesson details and objectives
Distinguish actual self-maps from plausible-looking axes.
- Determine rotations and reflections preserving labeled geometric shapes as sets.
- Differentiate rectangles, generic parallelograms and trapezoids by their symmetries.
- Justify completeness of regular-polygon symmetry lists.
6 practice tasks.
Construct and verify a motion witness
Lesson details and objectives
Give an executable sequence or a precise obstruction.
- Construct a sequence of rigid motions matching corresponding vertices.
- Verify the complete figure image rather than only a visual overlap.
- Use preserved distances and orientation to diagnose impossible or incomplete witnesses.
6 practice tasks.
Define triangle congruence through a witness
Lesson details and objectives
Connect side and angle correspondence to a whole-triangle rigid map.
- State triangle congruence using a rigid-motion composition and correspondence.
- Derive preserved corresponding sides and angles from that definition.
- Normalize a nondegenerate triangle by aligning one side and choosing its half-plane.
6 practice tasks.
Derive the three triangle criteria
Lesson details and objectives
Prove sufficient conditions by aligning a side and locating the third vertex.
- Derive SSS using distance equations and reflected third-point possibilities.
- Derive SAS from an included angle and ray-distance uniqueness.
- Derive ASA from intersecting rays, with half-plane and nondegeneracy conditions.
6 practice tasks.
Prove angle relations with parallel lines
Lesson details and objectives
Separate vertical-angle logic, Euclidean parallel-line facts and their converses.
- Prove vertical and transversal angle relations from stated Euclidean facts.
- Use converse angle criteria to establish parallelism.
- Derive triangle and exterior-angle sums with an auxiliary parallel.
6 practice tasks.
Track a dilation beyond the origin
Lesson details and objectives
Follow every point under a dilation and distinguish a line through the center from a parallel image.
- Calculate images about an arbitrary center using displacement vectors.
- Prove the line-image and segment-length rules for positive dilations.
- Explain why a dilation preserves angles but usually does not preserve distances.
6 practice tasks.
Construct a similarity and derive AA
Lesson details and objectives
Construct a similarity as motions plus a positive dilation, then derive AA by ray intersection.
- Build a pointwise similarity witness with correct correspondence.
- Derive AA without assuming similarity criteria.
- Use side ratios and angles to test or refute proposed similarities.
6 practice tasks.
Prove midpoint and median structure
Lesson details and objectives
Prove a midsegment theorem and the common 2:1 point of all medians.
- Derive the midpoint parallelism and half-length result.
- Prove all three medians meet at their 2:1 division point.
- Use congruence to justify an isosceles base-angle result.
6 practice tasks.
Characterize parallelograms and rectangles
Lesson details and objectives
Derive quadrilateral properties and keep the hypotheses of converse criteria.
- Prove opposite-side and opposite-angle properties of parallelograms.
- Prove the diagonal-bisection characterization.
- Justify an equal-diagonal rectangle criterion and reject it without the parallelogram hypothesis.
6 practice tasks.
Use proportional cuts and altitude similarity
Lesson details and objectives
Prove proportional cuts in both directions and recover Pythagoras from altitude similarities.
- Derive proportional segment cuts from parallel lines and AA.
- Prove the proportional-cut converse with an auxiliary parallel.
- Derive the right-triangle altitude identities and Pythagorean relation.
6 practice tasks.