Geometry — Free Practice Questions, Study Guides & Mock Exams

Learn the essentials of Geometry. In compliance with U.S. Public School Curriculum.

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Free Geometry study guides

Chapter 1.1: Points, Lines, and Planes — The starting vocabulary of geometry: the three undefined terms, the words built from them, how to name each figure correctly, and the basic postulates everything else will rest on.
Chapter 1.2: Linear Measure — Once we know what a segment is, we need to measure it. This topic gives you the two engines of all linear measurement — the Ruler Postulate and the Segment Addition Postulate — plus betweenness, congruent segments, and how to solve when the lengths are given algebraically.
Chapter 1.3: Distance and Midpoints — We carry the measuring tools of the number line into the coordinate plane. Two questions about any pair of points — where is the middle, and how far apart are they — are answered by the Midpoint Formula and the Distance Formula, the latter just the Pythagorean Theorem in disguise.
Chapter 1.4: Angle Measure — Angles record rotation and separation between rays. This topic defines the angle, the three ways to name it, the four basic types, the Protractor and Angle Addition Postulates, and the angle bisector — plus the algebra these set up.
Chapter 1.5: Angle Relationships — When angles meet, their measures often lock together. This topic catalogs the standard pairs — adjacent, linear, vertical, complementary, supplementary — plus perpendicular lines, and turns each into an equation you can solve.
Chapter 1.6: Two-Dimensional Figures — We put lengths and angles together to measure flat shapes. This topic defines polygons, classifies them by sides, and gives the two big questions about any region — perimeter (the boundary) and area (the surface inside) — with the formulas that answer them.
Chapter 2.1: Conjectures & Inductive Reasoning — How mathematicians discover ideas before they prove them: observing examples, spotting a pattern, stating a conjecture — and learning that a single counterexample can demolish it.
Chapter 2.2: Conditional Statements — The if-then sentence is the atom of every proof. Learn its parts, when it is true, and the four statements you can spin off it — converse, inverse, contrapositive, and the biconditional that defines.
Chapter 2.3: Deductive Reasoning — From guessing to guaranteeing. Where inductive reasoning only suggests, deductive reasoning proves — applying valid logical laws to accepted facts so the conclusion cannot be false.
Chapter 2.4: Postulates & Paragraph Proofs — What a proof is, the postulates you cite as reasons, and how to chain Given to Prove — one justified step at a time — in two-column and paragraph form.
Chapter 2.5: Algebraic Proofs — You can solve equations — but can you name the property behind every move? An algebraic proof turns equation-solving into a rigorous two-column argument where each step cites a Property of Equality.
Chapter 2.6: Proving Segment & Angle Relationships — From equality to congruence: the properties of congruence, the supplement and complement theorems, and the proof of the Vertical Angles Theorem you first met as a fact.
Chapter 3.1: Parallel Lines & Transversals — When a third line crosses two others, eight angles appear with a hidden structure. Learn to classify lines as parallel, skew, or intersecting, name the four transversal angle pairs, and use the Corresponding Angles Postulate to unlock every missing measure.
Chapter 3.2: Proving Lines Parallel — In 3.1 we started with parallel lines and read off the angles. Now we run the logic backwards: given the right angle evidence, we can conclude lines are parallel. These converses are your toolkit for proof.
Chapter 3.3: Slopes on the Coordinate Plane — Slope is the single number that captures a line's steepness and direction. Master the formula, the four slope types, and the slope tests for parallel and perpendicular lines, then use them to classify figures in the plane.
Chapter 3.4: Equations of Lines & Distance — Every non-vertical line has an equation — its precise address in the plane. Learn the two equation forms, write parallel and perpendicular lines through a chosen point, and measure perpendicular distances from a point to a line and between parallel lines.
Chapter 4.1: Classifying Triangles & Congruence — Triangles are the simplest, most rigid polygon — and the workhorse of every congruence proof to come. Here we sort them by sides and by angles, then build the language of congruent figures: correspondence, the congruence statement, and the principle called CPCTC.
Chapter 4.2: Angles of Triangles — Tear the corners off any paper triangle and they always line up flat. That single fact — the angles sum to $180^\circ$ — is the source of every angle theorem in this topic: the Angle-Sum Theorem, the Third Angles Theorem, and the Exterior Angle Theorem with its inequality.
Chapter 4.3: Triangle Congruence: SSS & SAS — Checking all six corresponding parts is tedious. The Side-Side-Side and Side-Angle-Side postulates let you prove two triangles congruent from just three pieces of information — provided you pick the right three. The catch lives in one word: included.
Chapter 4.4: Triangle Congruence: ASA & AAS — SSS and SAS let sides drive the proof. But diagrams often hand you angles more readily — through vertical angles, parallel lines, or givens. The Angle-Side-Angle postulate and the Angle-Angle-Side theorem let angles do the heavy lifting. The whole distinction comes down to where the side sits.
Chapter 4.5: Right Triangle Congruence (HL) — Right triangles are so common that they earn a congruence method of their own. The Hypotenuse-Leg theorem is the single valid "SSA-like" case — and it works for one precise reason: the Pythagorean Theorem locks in the missing leg the moment the hypotenuse and one leg are fixed.
Chapter 4.6: Isosceles & Equilateral Triangles — A triangle with two equal sides hides a built-in symmetry: those equal sides force equal angles. This topic proves the Isosceles Triangle Theorem and its converse, extends both to equilateral triangles, and turns them into fast angle-finding tools.
Chapter 5.1: Bisectors of Triangles — Two special points that live inside (or near) every triangle: the circumcenter, born from the perpendicular bisectors of the sides, and the incenter, born from the angle bisectors. Both come from one big idea — equidistance.
Chapter 5.2: Medians & Altitudes of Triangles — Two more special segments and the points they meet at: medians give the centroid — the triangle's balance point — and altitudes give the orthocenter. Plus the elegant Euler line that ties three centers together.
Chapter 5.3: Inequalities in One Triangle — Not every set of three lengths makes a triangle, and inside any triangle the angles and sides are ranked together. Three theorems govern this: the angle-side relationship, the exterior angle inequality, and the triangle inequality.
Chapter 5.4: Inequalities in Two Triangles — When two triangles share two pairs of equal sides but open to different angles, the wider one stretches its third side farther. That is the Hinge Theorem — and its converse runs the comparison backward, from sides to angles.
Chapter 6.1: Angles of Polygons — Before we study any one quadrilateral, we count the angles that every polygon must obey. Triangulation gives the interior-angle sum, a single full turn gives the exterior-angle sum, and regularity lets us divide each total evenly — the toolkit that powers every problem in this chapter.
Chapter 6.2: Properties of Parallelograms — A parallelogram is the parent of every special quadrilateral in this chapter. From its single defining feature — two pairs of parallel sides — flow four powerful theorems about its sides, angles, and diagonals that turn geometry problems into quick algebra.
Chapter 6.3: Proving Parallelograms — Topic 6.2 let us use a parallelogram's properties once we were told the shape was one. Now we run the logic backwards: given a few facts about a quadrilateral, which conditions force it to be a parallelogram? There are five reliable methods, and the art is choosing the one that fits your given information.
Chapter 6.4: Properties of Rectangles — A rectangle is a parallelogram with one extra demand: every angle is a right angle. That single condition gives it every parallelogram property for free, plus one new one of its own — congruent diagonals — and it opens the door to the Pythagorean theorem at every corner.
Chapter 6.5: Rhombi & Squares — A rhombus is a parallelogram with four equal sides, and that single condition gives its diagonals two striking powers a rectangle lacks: they meet at right angles and they bisect the corner angles. A square is the shape that is a rhombus and a rectangle at once — the most special quadrilateral of all.
Chapter 6.6: Trapezoids & Kites — Not every quadrilateral is a parallelogram. A trapezoid has just one pair of parallel sides; a kite has two pairs of adjacent congruent sides and no parallel pair at all. Each has its own special results — base angles, congruent diagonals, the midsegment average, and perpendicular diagonals — that round out the chapter.
Chapter 7.1: Dilations & Similarity Transformations — How a shape can grow or shrink without losing its shape: the center of dilation, the scale factor that controls everything, the coordinate rules, and the bigger idea of a similarity transformation that ties this whole chapter together.
Chapter 7.2: Similar Polygons & Ratios — When two polygons have the same shape: the precise two-part definition, how corresponding parts line up, how the scale factor lets you find missing sides, and how perimeter and area scale differently with that factor.
Chapter 7.3: Triangle Similarity: AA, SSS, SAS — Three shortcuts that confirm two triangles are similar without checking all six parts: Angle-Angle, Side-Side-Side, and Side-Angle-Side — plus how they power indirect measurement and the parallel-line setup.
Chapter 7.4: Proportional Parts & Parallel Lines — How parallel lines split sides in equal ratios: the Triangle Proportionality Theorem and its converse, the midsegment theorems for triangles and trapezoids, three-parallel-line transversals, and the angle bisector ratio.
Chapter 7.5: Special Segments & Geometric Mean — The altitude to the hypotenuse of a right triangle creates three similar triangles and an elegant chain of relationships — the geometric mean — that ties together the altitude, the legs, and the two hypotenuse segments.
Chapter 8.1: The Pythagorean Theorem — The most famous equation in geometry: how the two legs and the hypotenuse of a right triangle are bound together, how to find any missing side, the shortcut of Pythagorean triples, and how the converse classifies a triangle as right, acute, or obtuse.
Chapter 8.2: Special Right Triangles — Two right-triangle families — the 45-45-90 and the 30-60-90 — have fixed side ratios baked in. Learn the ratios, learn to spot them, and you can find every side from a single known length without ever touching the Pythagorean Theorem.
Chapter 8.3: Trigonometric Ratios — Sine, cosine, and tangent turn one angle and one side of a right triangle into every other measurement. Master SOH-CAH-TOA, learn to choose the right ratio, and use inverse trig to recover missing angles.
Chapter 8.4: Angles of Elevation & Depression — Turn a word problem about looking up or looking down into a right triangle, identify the angle and its sides, and apply SOH-CAH-TOA. The key fact: elevation and depression angles are equal, so you can always work from whichever vertex is easier.
Chapter 8.5: Law of Sines & Law of Cosines — SOH-CAH-TOA needs a right angle, but most real triangles have none. Two formulas — the Law of Sines and the Law of Cosines — solve any triangle. Learn which to reach for, and learn to handle the tricky ambiguous SSA case.
Chapter 9.1: Circles and Circumference — The vocabulary of the most symmetric shape in geometry — center, radius, diameter, chord, secant, tangent — and the four measurements that flow from it: circumference, area, arc length, and sector area.
Chapter 9.2: Arcs and Chords — How central angles measure arcs, how arcs add, and the tight web of "if-and-only-if" theorems linking congruent chords, congruent arcs, and equal distances from the center.
Chapter 9.3: Inscribed Angles — When an angle's vertex sits on the circle, it measures exactly half its intercepted arc. From that one theorem flow the same-arc corollary, the right-angle-in-a-semicircle rule, and the cyclic-quadrilateral law.
Chapter 9.4: Tangents and Secants — Lines that graze a circle or cut through it create their own precise rules: tangent perpendicular to radius, equal tangents from a point, the half-difference angle law, and the power-of-a-point segment products.
Chapter 9.5: Equations of Circles — The bridge from geometry to algebra: how the distance formula becomes $(x-h)^2+(y-k)^2=r^2$, how to read center and radius off an equation, complete the square, and test where a point lies.
Chapter 10.1: Areas of Triangles & Parallelograms — Area counts the unit squares that tile a flat figure. Here we build the base-times-height family of formulas — rectangle, square, parallelogram, and triangle — and learn the one rule that catches everyone: the height must be perpendicular to the base, never the slant side.
Chapter 10.2: Areas of Trapezoids, Rhombi & Kites — Three more quadrilaterals join the area family. A trapezoid averages its two parallel bases; a rhombus and a kite both split into half the rectangle their perpendicular diagonals span. Two formulas, three shapes — and one persistent trap involving the slant leg.
Chapter 10.3: Areas of Regular Polygons — A regular polygon has equal sides and equal angles. Slice it from the center into $n$ congruent triangles and a single formula falls out: $A=\tfrac12 aP$. The whole topic turns on one segment — the apothem — and one angle — the central angle.
Chapter 10.4: Areas of Circles & Sectors — The circle's area is $\pi r^2$; a sector is just a fraction $\theta/360^\circ$ of that pie. From those two facts come arc length, the circular segment (sector minus triangle), and the annulus — every curved region you will meet in this chapter.
Chapter 10.5: Areas of Composite Figures — Real shapes — floor plans, gardens, metal brackets — are rarely a single rectangle. A composite figure is built from basic shapes you already know. The plan never changes: decompose, calculate each piece, then add or subtract.
Chapter 11.1: 3D Solids & Nets — Stepping out of the flat plane: the vocabulary of three-dimensional solids, the families of polyhedra, Euler's counting law, and the flat "nets" that fold up into the solids whose surface area you will measure for the rest of this chapter.
Chapter 11.2: Surface Area: Prisms & Cylinders — How much material wraps a box or a can? By unrolling the lateral surface into a single rectangle, every prism and cylinder reduces to the same idea — perimeter times height, plus the two bases.
Chapter 11.3: Surface Area: Pyramids & Cones — Pointed solids taper to a single apex, and their lateral faces are measured with a new length — the slant height. Get the slant height right and every pyramid and cone surface-area formula falls into place.
Chapter 11.4: Volume of Solids — How much space does a solid enclose? Two ideas carry the whole topic: prisms and cylinders are base-area times height, and pyramids and cones are exactly one-third of their matching prism or cylinder.
Chapter 11.5: Spheres: Surface Area & Volume — The most symmetric solid of all. Two clean formulas — $4\pi r^2$ for surface area and $\tfrac{4}{3}\pi r^3$ for volume — plus hemispheres and the elegant sphere-in-cylinder ratio that Archimedes prized above all his results.
Chapter 12.1: Sample Spaces & Counting Principles — Before you can measure how likely something is, you have to list everything that could happen — and count it without missing a case. This topic builds the sample space, the Fundamental Counting Principle, and the difference between theoretical and experimental probability.
Chapter 12.2: Permutations & Combinations — Two counting questions look almost identical but give wildly different answers: how many ways to arrange items versus how many ways to choose them. The deciding question is always the same — does order matter?
Chapter 12.3: Geometric Probability — When outcomes are not countable points but a smooth length, area, or angle, probability becomes a ratio of measures. A randomly thrown dart or a spun arrow has infinitely many landing spots — so we compare sizes, not counts.
Chapter 12.4: Independent & Dependent Events — When two things both have to happen, you multiply — but exactly what you multiply depends on whether the first event changes the second. This topic builds the multiplication rule, conditional probability, and the all-important "with or without replacement" check.
Chapter 12.5: Frequency Tables & Decision Making — Real data rarely arrives as tidy fractions — it arrives in counts. A two-way frequency table organizes those counts so you can read off probabilities, compare groups, detect association, and weigh choices with expected value.

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Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12

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