Geometry

Shape, proof and measurement, built from what each argument can justify

About this course

Geometric reasoning, construction and measurement — proof from parallels and rigid motions, triangle congruence and similarity, circles, coordinate proof, area and solids.

What you will be able to do

  • Write a geometric proof using parallel-line angle facts and rigid motions
  • Decide which measurements determine a triangle, and carry out constructions and loci
  • Use similarity and the Pythagorean theorem to measure inaccessible lengths
  • Prove and apply angle, tangent, chord and secant results for circles
  • Choose coordinates to prove a result, and derive area formulas and properties of solids

What this course covers

  • Choosing coordinate axes without adding assumptions
  • Triangle midpoint theorem via coordinate proof
  • Coordinate-free midpoint vector form
  • Division-free tests for parallel and perpendicular directions
  • Varignon parallelogram: midpoints of a quadrilateral's sides
  • Special Varignon parallelograms from diagonal properties
  • Translating between geometry and coordinates: habits and pitfalls
  • Inscribed angles and intercepted arcs: definitions
  • Inscribed angle theorem: half the intercepted arc
  • Proof via isosceles triangles and a diameter (base case)
  • Proof extension: center inside, on boundary, or outside the inscribed angle
  • Angles in the same segment and the semicircle right angle
  • Cyclic quadrilaterals: opposite angles are supplementary
  • Converse: supplementary opposite angles certify cyclicity
  • Givens, conclusions, and appearances distinguished
  • Counterexamples and universal claims
  • Examples versus proof for unlimited families
  • Vertically opposite angles theorem
  • Converse of a geometric statement
  • Repairing a proof by replacing appearance-based reasons
  • Structure of a dependable geometric argument
  • Compass and straight-edge as logical tools
  • Perpendicular bisector construction
  • Locus of points equidistant from two fixed points
  • Perpendicular from an external point (application)
  • Angle bisector construction and its locus
  • Intersecting loci and region restrictions
  • Segment solution and precision of language
  • Dilation: center, factor, and point mapping
  • Angles and lines preserved under dilation
  • Center choice, placement, and similarity definition
  • Triangle similarity criteria
  • Area scales as the square of the length factor
  • Volume scales as the cube of the length factor
  • Applying scale laws: recovering factors and classifying quantities
  • Area vs. perimeter distinction and unit squares
  • Dissection and rigid-motion area preservation
  • Perpendicular height vs. sloping side
  • Triangle and trapezium area formulas from dissection
  • Composite regions with inward corners
  • Similarity and the squared area scale factor
  • Circle area via sector refinement and exhaustion
  • Four triangle centers: overview and locus tools
  • Centroid from medians: two-to-one ratio and concurrency
  • Circumcenter from perpendicular bisectors: equal vertex distances
  • Incenter from internal angle bisectors: equal side distances
  • Orthocenter from altitudes: concurrency and exterior position
  • Equilateral triangle: coincidence of all four centers
  • Selecting a center from a practical requirement
  • Statement and Geometric Meaning of the Theorem
  • Foundations and Circular-Reasoning Pitfall
  • Dissection Proof via Four Congruent Triangles
  • Conservation-of-Area Proof via Two Rearrangements
  • Similarity Proof via Altitude to the Hypotenuse
  • The Converse of the Pythagorean Theorem
  • Synthesis: Three Proofs, Common Errors, and Proof Habits
  • Applying the Theorem and Converse to Calculations
  • Angles at a single intersection
  • Corresponding, alternate interior, and same-side interior angle pairs
  • Parallel angle principle and its converse
  • Algebraic angle problems with parallel lines
  • Triangle angle sum theorem
  • Impossible triangle data diagnosis
  • Exterior angle theorem
  • Tangents, chords, and secants: definitions and setup
  • Perpendicular from center bisects a chord
  • Tangent perpendicularity and equal tangent lengths
  • Tangent–secant power relationship
  • Two-secant power theorem and the power of a point
  • Intersecting chords inside a circle
  • Capstone worked example and decision procedure
  • Similarity cancels scale: ratios depend on angle
  • Defining sine, cosine, and tangent
  • Effect of moving the reference angle
  • Exact values for 30°, 45°, and 60°
  • Translating a sightline into tower height
  • Sloping ground: separating horizontal run from slope distance
  • Exact model results versus input-range bounds
  • Congruence via rigid motions: definition and setup
  • Translation: shared displacement and why it preserves the figure
  • Congruence correspondence and vertex labeling
  • Rotation: center, angle, direction, and unequal travel distances
  • Reflection: perpendicular bisection, orientation reversal
  • Composition of rigid motions: order matters
  • Motion certificate: supplying and verifying a congruence proof
  • Nets, surface area, and open containers
  • Representations: nets, perspective views, and orthogonal views
  • Cross-sections of prisms and pyramids
  • Cavalieri's principle and the pyramid volume formula
  • Cylinder and cone: perpendicular height vs. slant length
  • Sphere volume via the cylinder-minus-cone section comparison
  • Scaling: lengths, areas, and volumes under uniform similarity
  • Convex quadrilateral fundamentals and angle sum
  • Parallelogram equivalence: parallel sides and bisecting diagonals
  • Rectangle characterization: equal diagonals within a parallelogram
  • Inclusive hierarchy and the square's dual membership
  • Rhombus characterization: perpendicular diagonals within a parallelogram
  • Square as intersection of rectangle and rhombus families
  • Worked classification and side-length calculation using diagonal conditions
  • What 'determines' a triangle means
  • SSS — three sides determine a triangle
  • Triangle inequality as existence condition
  • SAS — two sides and included angle determine a triangle
  • ASA and AAS — angle-side combinations determine a triangle
  • SSA — failure as a general congruence criterion
  • Decision process and summary of congruence criteria
  • Prerequisites: scale, area, volume, and perpendicular distance
  • Path-width clearance proof using region extremes
  • Translating a design brief into regions and constraints
  • Locating a point by intersecting two loci
  • Area partition and paving volume calculation
  • Maximum radius via combined inequalities
  • Assembling a defensible proof portfolio

Syllabus

  1. Proof, Parallels and Congruence

    1. L1 · What Makes a Geometric Argument a Proof
    2. L2 · What Parallel Lines Force
    3. L3 · Congruence Through Rigid Motions
    4. L4 · Which Measurements Determine a Triangle

  2. Constructions and Triangle Centres

    1. L5 · Constructions and Loci
    2. L6 · The Centres of a Triangle

  3. Similarity and Measurement

    1. L7 · What Similarity Preserves
    2. L8 · Why the Pythagorean Theorem Is True
    3. L9 · Measuring an Inaccessible Length
    4. L10 · The Family of Quadrilaterals

  4. Circles

    1. L11 · Angles in a Circle
    2. L12 · Tangents, Chords and Secants

  5. Coordinates, Area and Solids

    1. L13 · Choosing Coordinates for a Proof
    2. L14 · How Area Formulas Fit Together
    3. L15 · From Flat Drawings to Solids
    4. L16 · A Design That Can Be Justified