Grade 10 · printable course

Geometry: Structure, Proof, and Measurement

Grade 10 · 565 pages · 74 lessons · PDF, print at home

A full-year course building a logical system for reasoning about shape, space, and measurement, grounded in transformations as the definition of congruence and similarity. Students move from informal observation to formal proof, apply coordinate and algebraic tools to geometric objects, and extend triangle relationships into right-triangle trigonometry and circle theorems. The course closes with three-dimensional measurement and an introduction to geometric probability, tying the year's proof and measurement strands together. Throughout, students learn to distinguish a plausible diagram from a proven claim.

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Every unit and lesson in this book, in order. The same lessons are taught online at lettersanddevotion.com, so a family can move between paper and screen.

Geometry: Structure, Proof, and Measurement

74 lessons · try a lesson online free

Unit 1: Transformations and the Definition of Congruence/Similarity8 lessons

1.Coordinate rules for translation, reflection, and rotation
2.Identification criteria distinguishing translation, reflection, and rotation
3.The definition of congruence as existence of a rigid motion
4.The coordinate dilation rule (multiply coordinates relative to center by scale factor)
5.Preserved vs. non-preserved properties across rigid motions and dilations
6.The definition of similarity as dilation composed with rigid motion
7.Application of the rigid-motion/dilation distinction outside a coordinate-grid context
8.Generalization of the preserved-property testing strategy to an unnamed, untaught transformation

Unit 2: Triangle Congruence and the Logic of Proof10 lessons

9.Correspondence of parts under a rigid motion mapping
10.The logical sufficiency of SSS/SAS/ASA for guaranteeing a congruence-producing rigid motion
11.SSA and AAA as insufficient congruence criteria
12.Triangle congruence criteria (SSS, SAS, ASA, AAS) applied to diagrams requiring an inferred part
13.Two-column proof structure applied to vertical angles and alternate interior angles theorems
14.Triangle congruence proof for overlapping-triangle diagrams, including a CPCTC-based conclusion
15.The isosceles triangle theorem, proved via an auxiliary bisector and SAS/CPCTC
16.Selecting a minimal sufficient measurement set to certify congruence in an unfamiliar physical/design context
17.Diagnosing why a specific SSA claim fails, in a novel diagram
18.Planning a proof strategy for an untaught geometric claim by identifying which congruence criteria would apply

Unit 3: Similarity and Proportional Reasoning in Triangles9 lessons

19.The AA similarity criterion for triangles
20.The AA, SAS~, and SSS~ similarity criteria as a decision set
21.The asymmetry between minimum evidence needed for similarity versus congruence
22.The triangle proportionality theorem (a line parallel to one side divides the other two sides proportionally)
23.The triangle proportionality theorem and its converse, and their shared logical dependence on AA similarity
24.The geometric mean relationships produced by the altitude-to-the-hypotenuse configuration
25.A similarity-based proof of the Pythagorean theorem
26.The similar-triangle structure underlying shadow, mirror, and scale-drawing measurement methods
27.The combined decision space of congruence criteria (Unit 2) and similarity criteria (this unit)

Unit 4: Coordinate Geometry: Proof and Measurement on the Plane10 lessons

28.The distance formula as a Pythagorean-theorem application
29.Slope criteria for parallel and perpendicular lines
30.Midpoint formula and ratio-based segment partition
31.The logical link between slope/distance evidence and the definition of a parallelogram
32.Coordinate proof of rectangle/rhombus properties
33.Strategic placement of a figure on coordinate axes
34.Original coordinate proof construction from a verbal description
35.The gap between visual appearance on a grid and a computed coordinate proof
36.The relationship between synthetic proof and coordinate proof of the same geometric fact
37.Distance formula, midpoint formula, slope criteria for parallel/perpendicular

Unit 5: Right Triangle Trigonometry9 lessons

38.Invariance of the opposite/hypotenuse ratio across similar right triangles
39.Definitions of sine, cosine, and tangent as side ratios
40.Solving for an unknown side using a selected trig ratio
41.Using inverse trigonometric functions to find an angle measure
42.Diagramming angle of elevation and angle of depression problems
43.The complementary angle identity sin(theta) = cos(90-theta)
44.Justifying ratio choice by naming the similar-triangle relationship in an indirect-measurement context
45.Extending right-triangle trig to a non-right-triangle context via an constructed altitude
46.Equivalence of the similarity-proportion method and the trig-ratio method for indirect measurement

Unit 6: Circles: Angles, Arcs, Segments, and Equations11 lessons

47.Congruence of radii within a single circle and the resulting isosceles triangle
48.The inscribed angle theorem and its proof via isosceles triangles formed by radii
49.Congruence of two tangent segments from an external point, proved via HL/SSS triangle congruence
50.The chord-chord segment product relationship and its derivation from AA similarity
51.Discrimination among chord-chord, secant-secant, and secant-tangent segment relationships
52.Arc length and sector area as proportional fractions of circumference and circle area
53.The circle equation as an instance of the distance formula applied to a fixed center and radius
54.Completing the square to convert a circle equation from general to standard form
55.Discrimination among distance, slope, and circle-equation methods on the coordinate plane
56.Composite area of a circle-and-inscribed-polygon figure, combining multiple unit strands in an untaught configuration
57.Validity of justification lines in a circle-theorem proof, distinguishing diagram-based claims from criterion-based claims

Unit 7: Three-Dimensional Measurement8 lessons

58.Cavalieri's principle (equal cross-sectional area at every height implies equal volume)
59.Volume formulas for prisms, cylinders, pyramids, and cones
60.Surface area via net decomposition
61.The sphere volume formula derived via Cavalieri's principle (hemisphere vs. cylinder-minus-cone)
62.The effect of scale factor k on length (k), area (k^2), and volume (k^3)
63.Volume and surface area of composite solids modeling a real object
64.Density as a conversion between volume and mass for real-world objects
65.Matching real-world objects to appropriate solid models (including frustums and partial solids)

Unit 8: Geometric Probability and Modeling with Measurement9 lessons

66.The distinction between discrete (countable) and continuous (measurable) sample spaces
67.Length-based geometric probability ratio on a one-dimensional interval
68.Single-region area-based geometric probability ratio, reproducing the exact worked-example structure
69.Area-based geometric probability ratio for composite and nested regions requiring subtraction before dividing
70.Dimensional consistency requirement in geometric probability ratios
71.The uniform-distribution assumption underlying geometric probability models and conditions under which it fails
72.A complete geometric probability model for an authentic scenario, including its ratio, computation, and stated assumption
73.Density as a ratio of count to area/volume measure, and its use in estimating counts across regions
74.The structural distinction between geometric and classical probability models as applied to the same random process

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