For evaluators and planning parents

Standards Alignment

Every one of the 48 propositions in Euclid's Elements Book I, mapped to the Common Core State Standards for Geometry. Three clusters are covered thoroughly. Three are not covered at all, and those are listed here with the same prominence as the rest.

Filling the gaps takes roughly 12 to 16 weeks of supplementary work on top of the 36-week Euclid year. Most families spread that across the year rather than stacking it at the end.

What Book I covers

G-CO.D

Covered

Make geometric constructions

Every compass-and-straightedge construction the standard lists is in Book I, and Euclid proves each one works rather than presenting it as a procedure. Constructing a regular hexagon inscribed in a circle is the one exception; it appears in Book IV.

Propositions: 1, 2, 3, 9, 10, 11, 12, 22, 23, 31, 46

G-CO.C

Covered

Prove geometric theorems

The deepest coverage in Book I. Vertical angles (Prop 15), the parallel-line angle relationships (Props 27 to 29), the triangle angle sum (Prop 32), and the parallelogram properties (Prop 34) are all proved outright, alongside a great deal the standard never asks for.

Propositions: 5, 6, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45

G-CO.B

Covered

Understand congruence in terms of rigid motions

SAS (Prop 4), SSS (Prop 8), and ASA with AAS (Prop 26) are proved from first principles. Euclid argues by superposition rather than in the modern language of rigid motions, so connect the two explicitly for your student.

Propositions: 4, 8, 26

What it covers partly

G-SRT.B

Partly covered

Prove theorems involving similarity

The Pythagorean Theorem (Prop 47) and its converse (Prop 48) are here, proved by areas rather than by similarity as the standard envisions. Similar triangles and proportional reasoning are in Books V and VI, not Book I.

Propositions: 47, 48

To fill this gap: 3 to 4 weeks. Continue into Euclid's Books V and VI to stay with the primary source, or use a modern treatment of similar triangles.

What it does not cover

G-CO.A

Not covered

Experiment with transformations in the plane

The largest gap. Translations, rotations, and reflections as functions on the plane are a modern idea Euclid never developed. His definitions (D1 to D23) supply every primitive those definitions are built from, and nothing about the transformations themselves.

To fill this gap: 2 to 3 weeks. A transformational geometry unit. GeoGebra is free and handles this well.

G-GPE

Not covered

Expressing geometric properties with equations

Entirely outside the Elements. Descartes introduced coordinates around 1637, roughly nineteen centuries after Euclid. There is no distance formula, midpoint formula, slope, or equation of a line or circle anywhere in Book I.

To fill this gap: 3 to 4 weeks. A coordinate geometry unit covering the Cartesian plane, distance and midpoint, slope, and equations of lines and circles.

G-C

Not covered

Circles

Book I uses circles as construction tools and proves no theorems about them. Inscribed angles and tangent lines are in Book III.

To fill this gap: 2 to 3 weeks. Euclid's Book III continues the primary-source approach, or use a modern circles unit.

G-GMD

Not covered

Geometric measurement and dimension

Volume, cross-sections, and Cavalieri's principle are not in Book I.

To fill this gap: 2 to 3 weeks. A three-dimensional geometry and measurement unit.

G-MG

Not covered

Modeling with geometry

Applied geometric modeling is not addressed in Book I.

To fill this gap: ongoing. Real-world application projects folded in across the year rather than taught as a unit.

Theorems Book I implies without stating

These sit inside clusters Euclid otherwise covers well. Each takes a lesson or two to close.

Standard	Theorem	Where it sits	Effort
G-CO.C.9	Perpendicular bisector characterization	Props 5, 6, and 12 hold every piece; Euclid never states the biconditional.	One synthesis lesson
G-CO.C.10	Midsegment theorem	Derivable from Props 29 to 31.	One or two lessons
G-CO.C.10	Medians meet at a point	Outside Book I's technique entirely.	Supplement with an area or coordinate proof
G-CO.C.11	Diagonals of a parallelogram bisect each other	Follows from Prop 34 with Prop 4, never stated.	One lesson
G-CO.D.13	Regular hexagon inscribed in a circle	Book IV, Prop 15, not Book I.	One lesson

Proposition by proposition

Every proposition links to its page on this site, where the full statement, diagram, and proof are free to read.

Proposition	Statement	Type	Standards
Proposition 1	Construct an equilateral triangle on a line	Construction	G-CO.D.12, G-CO.D.13
Proposition 2	Copy a segment to a new point	Construction	G-CO.D.12
Proposition 3	Cut off a segment equal to a shorter one	Construction	G-CO.D.12
Proposition 4	SAS triangle congruence	Theorem	G-CO.B.6, G-CO.B.7, G-CO.B.8
Proposition 5	Base angles of an isosceles triangle are equal	Theorem	G-CO.C.9, G-CO.C.10
Proposition 6	Equal angles imply equal opposite sides	Theorem	G-CO.C.10
Proposition 7	Uniqueness of triangle construction on a base	Theorem	G-CO.B.8
Proposition 8	SSS triangle congruence	Theorem	G-CO.B.6, G-CO.B.7, G-CO.B.8
Proposition 9	Bisect an angle	Construction	G-CO.D.12
Proposition 10	Bisect a line segment	Construction	G-CO.D.12
Proposition 11	Perpendicular from a point on a line	Construction	G-CO.D.12
Proposition 12	Perpendicular from a point not on a line	Construction	G-CO.D.12
Proposition 13	Angles on a line sum to two right angles	Theorem	G-CO.C.9
Proposition 14	Supplementary adjacent angles form a line	Theorem	G-CO.C.9
Proposition 15	Vertical angles are equal	Theorem	G-CO.C.9
Proposition 16	Exterior angle exceeds either remote interior angle	Theorem	G-CO.C.9, G-CO.C.10
Proposition 17	Two angles of a triangle sum to less than 180 degrees	Theorem	G-CO.C.10
Proposition 18	Greater side opposite greater angle	Theorem	G-CO.C.10
Proposition 19	Greater angle opposite greater side	Theorem	G-CO.C.10
Proposition 20	Triangle inequality	Theorem	G-CO.C.10
Proposition 21	Interior lines are shorter but contain a greater angle	Theorem	G-CO.C.10
Proposition 22	Construct a triangle from three given segments	Construction	G-CO.D.12
Proposition 23	Copy an angle to a new location	Construction	G-CO.D.12
Proposition 24	Hinge theorem (SAS inequality)	Theorem	G-CO.C.10
Proposition 25	Converse of the hinge theorem	Theorem	G-CO.C.10
Proposition 26	ASA and AAS triangle congruence	Theorem	G-CO.B.7, G-CO.B.8
Proposition 27	Equal alternate angles imply parallel lines	Theorem	G-CO.C.9
Proposition 28	Equal corresponding angles imply parallel lines	Theorem	G-CO.C.9
Proposition 29	Parallel lines and the angles a transversal makes	Theorem	G-CO.C.9
Proposition 30	Transitivity of parallelism	Theorem	G-CO.C.9
Proposition 31	Construct a parallel through a given point	Construction	G-CO.D.12
Proposition 32	Angles of a triangle sum to two right angles	Theorem	G-CO.C.10
Proposition 33	Joining ends of equal parallel segments gives a parallelogram	Theorem	G-CO.C.11
Proposition 34	Opposite sides and angles of a parallelogram are equal	Theorem	G-CO.C.11
Proposition 35	Parallelograms on the same base between the same parallels are equal	Theorem	G-CO.C.11
Proposition 36	Parallelograms on equal bases between the same parallels are equal	Theorem	G-CO.C.11
Proposition 37	Triangles on the same base between the same parallels are equal	Theorem	G-CO.C.11
Proposition 38	Triangles on equal bases between the same parallels are equal	Theorem	G-CO.C.11
Proposition 39	Equal triangles on the same base lie between the same parallels	Theorem	G-CO.C.11
Proposition 40	Equal triangles on equal bases lie between the same parallels	Theorem	G-CO.C.11
Proposition 41	A parallelogram is double the triangle on the same base	Theorem	G-CO.C.11
Proposition 42	Construct a parallelogram equal to a triangle in a given angle	Construction	G-CO.C.11
Proposition 43	Complements about the diagonal of a parallelogram are equal	Theorem	G-CO.C.11
Proposition 44	Apply a parallelogram equal to a triangle on a given line	Construction	G-CO.C.11
Proposition 45	Construct a parallelogram equal to any rectilinear figure	Construction	G-CO.C.11
Proposition 46	Construct a square on a given line	Construction	G-CO.D.12, G-CO.D.13
Proposition 47	The Pythagorean Theorem	Theorem	G-SRT.B.4
Proposition 48	Converse of the Pythagorean Theorem	Theorem	G-SRT.B.4

Classical education

Book I is the foundational text for the Geometry arm of the Quadrivium, a position it has held for more than two millennia. It sits in the logic stage of the Trivium, roughly ages 12 to 16, because every proposition asks a student to move from stated premises to a necessary conclusion using only what has already been proved.

For a classical education, the clusters listed above as gaps are gaps only against one particular standard. Many classical educators defer coordinate geometry, transformations, and similarity deliberately, on the view that a student should develop real facility with axiomatic reasoning first. On those terms Book I is a complete, self-contained course.

Using this page

If you need to show coverage to an evaluator, charter administrator, or umbrella school, this page is public and linkable. Nothing here claims accreditation, and no standards body has reviewed this curriculum. It is our own mapping, and every row can be checked against the proposition it names.

Questions about a specific standard? Email hello@euclidsworkshop.com. The full interactive course covering all 48 propositions is free to use, and the printable curriculum adds lesson plans, worksheets, and answer keys.