How to Use This Book

Geometry · 2023 Virginia Standards of Learning

This book covers every Geometry mathematics standard Virginia expects students to learn, in a sequence built for teaching rather than for filing. It is free to read, print, and share.

For students

Geometry asks a question the courses before it mostly did not: how do you know? In Algebra 1, an answer was a number, and checking it meant substituting. Here an answer is often a claim about a figure — these triangles are congruent, that quadrilateral is a rhombus, this angle measures $58°$ — and checking it means producing a reason someone else can follow. That is what a proof is. It is not a special kind of problem; it is what a complete answer looks like in this course.

Each chapter opens by telling you exactly what you will be able to do when you finish it. Work through a lesson in this order:

Read the narrative. New words appear in bold the first time they are used.
Study the figure before the sentence about it. In this book the picture usually carries the given information, and the sentence tells you what to do with it. Marks matter: a single tick and a double tick on two segments say those pairs are congruent to each other, not to anything else, and a square corner means exactly $90°$ .
Follow the worked examples. Every step is shown, with the reason beside it. Cover the reason column, try to supply it yourself, then check.
Do the guided practice. These are scaffolded — the earlier ones give you more help than the later ones.
Do the independent practice. Each set includes at least one real-world application and one reasoning problem where you explain your thinking. In Geometry, the explaining is most of the subject.
Take the exit ticket. A few questions. If you miss one, go back to the matching part of the lesson before moving on.

Answers to everything are in the answer key. Use them to check your work after you try a problem, not before.

How the problems are numbered

In this book, item numbers run straight through a whole chapter. Lesson 1's guided practice starts at item 1, and the numbers keep climbing to the last item of the chapter review — they do not restart at each lesson. So "item 47" names exactly one problem in the chapter, which is what lets the answer key and the coverage table point at specific problems without ambiguity.

Notation this book holds to

Congruent is not equal. Segments and angles are congruent; their measures are equal. Write $\overline{AB} \cong \overline{CD}$ or $AB = CD$ , and $\angle A \cong \angle B$ or $m\angle A = m\angle B$ — never a mix of the two in one statement.
Order carries meaning. $\triangle ABC \cong \triangle DEF$ says $A$ corresponds to $D$ , $B$ to $E$ , and $C$ to $F$ . Reordering the letters makes a different claim, so the answer key treats a correct pair of triangles named in the wrong order as an incomplete answer and says why.
Exact first, rounded second. A side of $6\sqrt{2}$ is the answer; $8.49$ is that answer rounded. Give the exact value, then a decimal if the problem asks for a measurement, labelled about. Areas and volumes are given in terms of $\pi$ before they are given as decimals.
Arc measure and arc length are different quantities. An arc's measure is in degrees and matches its central angle. An arc's length is a distance, in the same units as the radius. Chapter 14 keeps them apart on purpose.
Every angle in this book is measured in degrees.

About the calculator

Geometry is an End-of-Course subject, and VDOE provides the Desmos Virginia Graphing Calculator for the entire test. There are no no-calculator Geometry standards, and VDOE supplies a formula sheet with the test as well.

Read those two facts together and you can see what this course actually grades. It is not arithmetic, and it is not whether you memorized the volume of a cone. It is whether you chose the right relationship and can say why it applies. So this book asks for the reason before the number, every time:

Name the postulate, the angle pair, or the similarity criterion first.
Then compute — by hand or with the calculator, whichever is faster.
When a computed answer disagrees with the figure, trust neither until you find the disagreement. A tangent ratio that produces a hypotenuse shorter than a leg is telling you something.

Chapter 9 is the one place where the calculator is genuinely required, because sine, cosine, and tangent values are not constructible by hand. There this book rounds side lengths to the nearest hundredth and angle measures to the nearest degree, computes from unrounded intermediate values, and states the rounding in every answer.

For teachers

Sequence. VDOE presents the standards grouped by content strand — Reasoning, Lines, and Transformations; Triangles; Polygons and Circles; Three-Dimensional Figures — and leaves instructional sequence to local curricula. This book follows the strand order, which in Geometry is genuinely close to a teaching order: the vocabulary of proof has to exist before triangles are proved congruent, congruent triangles are what the standard proofs of quadrilateral properties use, and plane figures are what solids are built from. Chapters are self-contained enough to reorder, with these dependencies worth respecting:

Logic and conditional statements (Chapter 1) come before every chapter that says "prove."
Parallel lines and transversals (2) come before triangle angle relationships (4) and before the parallelogram proofs (10).
Transformations (3) come before congruence (5–6) and similarity (7): congruence is what rigid motions preserve, and similarity is what dilation adds.
Congruent triangles (5–6) come before quadrilateral properties (10–11).
Similar triangles (7) come before right-triangle trigonometry (9), which is well defined only because same-angle right triangles are similar.
The Pythagorean Theorem (8) comes before the equation of a circle (15), which is derived from it, and before slant heights in surface area (16).
Surface area and volume (16) come before changing dimensions (17).

Chapter-to-standard mapping. Every chapter closes with a coverage table showing which Knowledge and Skill letter is taught in which lesson and practiced in which items. The volume's SOL coverage index lists all thirteen standards and where each is addressed.

Where a standard is split. Four standards are taught across more than one chapter, because their Knowledge and Skills bundle distinct skills:

G.TR.2 — synthetic proof and constructions (Chapter 5); algebraic and coordinate proof (6)
G.TR.4 — the Pythagorean Theorem, its converse, and the special right triangles (Chapter 8); the trigonometric ratios and angles of elevation and depression (9)
G.PC.1 — properties and deductive proof (Chapter 10); coordinate and algebraic proof (11)
G.PC.3 — central and inscribed angles (Chapter 13); arc length and sector area (14)

Two bullets are deliberately taught twice: G.TR.2e, because problems about the measured attributes of congruent triangles are the payoff of a congruence proof by either method, and G.PC.1a, because the property list of Chapter 10 is exactly what the coordinate computations of Chapter 11 are testing for.

What the 2023 standards do not ask for. Geometry courses have historically carried more than the standards require, and this book does not. In particular: the circle standard G.PC.3 names central angles, inscribed angles, arc length, and sector area, and does not name chord-chord, secant-secant, or tangent-secant angle and segment relationships, so this book does not teach them. G.PC.4 names the standard form of a circle's equation and four problem types built on it, and does not name completing the square to recover a center from general form. G.TR.4 names sine, cosine, and tangent, and does not name the law of sines or the law of cosines. G.PC.2 says convex polygons. Where a teacher's local curriculum adds any of this back, it is enrichment, and it should be labelled that way to students.

Constructions are assessed. Both G.TR.2d and G.PC.1d name compass-and-straightedge constructions as a way of verifying a property. This book draws the construction arcs and asks students to keep theirs, because the arcs are the justification. A construction with the arcs erased is a drawing.

Assessment. Each lesson has an exit ticket. Every chapter has an end-of-chapter review organized by Knowledge and Skill, so a review score points at a specific standard rather than a vague topic.

What builds into Geometry, and what Geometry builds toward

Grade 8 supplied the angle-pair relationships, translations and reflections in the coordinate plane, and the Pythagorean Theorem. Algebra 1 supplied slope, the distance between two points as a computation, systems, and the habit of verifying a result a second way. Geometry takes all of that and changes what counts as an answer. Four shifts define the course:

Justification becomes the product. A number with no reason is incomplete. Direct proofs, indirect proofs, and coordinate proofs are three tools for the same job, and the standards name all three.
Transformations become a definition, not a trick. Two figures are congruent when a sequence of rigid motions carries one onto the other, and similar when a dilation is allowed as well. That is why congruence criteria work and why trigonometric ratios exist.
Algebra becomes a way to prove. Slope, distance, and midpoint turn a claim about a figure into a computation in the coordinate plane — which is the bridge from Algebra 1 and the reason Chapters 6, 11, and 15 exist.
Two dimensions become three. Surface area, volume, cross sections, and the effect of scaling a dimension close the course, and the scaling result — lengths by $k$ , areas by $k^2$ , volumes by $k^3$ — is the single most transferable fact in it.

By the end of the course, a student should be able to write a two-column or paragraph proof that two triangles are congruent or similar and say which criterion carried it; find an unknown length or angle in a figure and name the relationship that justified it; work in the coordinate plane with slope, distance, and midpoint to prove a quadrilateral is what it looks like; solve a right triangle with trigonometry to a stated rounding; and compute the surface area or volume of a solid, then predict what happens to both when a dimension changes.

Sources and independence

Standards text is drawn from the Mathematics Standards of Learning for Virginia Public Schools, 2023, approved by the Virginia Board of Education on August 31, 2023 and fully implemented in the 2024–2025 school year. The Geometry calculator policy described above is drawn from VDOE's Desmos Online Calculator guidance for End-of-Course mathematics tests.

This book is an independent open educational resource. It is not published, reviewed, or endorsed by the Virginia Department of Education. Where this book says a standard requires something, the requirement comes from the standard's own Knowledge and Skills text; the teaching sequence, examples, exercises, and explanations are original to this book.