Geometry Cheat Sheet

The core ideas of Geometry distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Euclidean Geometry

The study of geometry based on Euclid's five postulates, dealing with flat (planar) space and three-dimensional space. It forms the foundation of most high-school geometry, covering congruence, similarity, parallelism, and the properties of common shapes like triangles, circles, and polygons.

Pythagorean Theorem

A fundamental relation in Euclidean geometry stating that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides ($a^2 + b^2 = c^2$). It connects algebra and geometry and serves as the basis for the distance formula in coordinate geometry.

Congruence and Similarity

Two figures are congruent if they have the same shape and size, and similar if they have the same shape but possibly different sizes. Congruence criteria (SSS, SAS, ASA, AAS) and similarity criteria (AA, SAS, SSS) are essential tools for proving geometric relationships.

Coordinate Geometry (Analytic Geometry)

A method of studying geometry by placing figures on a coordinate plane and using algebraic equations to describe them. This approach allows geometric problems to be solved with algebraic techniques such as finding distances, midpoints, and slopes.

Transformations

Operations that move or change geometric figures while preserving certain properties. The four main rigid transformations (isometries) are translation, rotation, reflection, and glide reflection, each preserving distance and angle measure. Dilations change size but preserve shape.

Circle Theorems

A collection of results about angles, arcs, chords, tangents, and secants related to circles. Key theorems include the inscribed angle theorem, the tangent-radius perpendicularity theorem, and the power of a point theorem, all of which are central to Euclidean geometry proofs.

Area and Volume Formulas

Systematic methods for computing the measure of two-dimensional regions (area) and three-dimensional solids (volume). These formulas range from simple expressions like base times height for parallelograms to integral-based approaches for irregular shapes.

Non-Euclidean Geometry

Geometric systems in which Euclid's parallel postulate does not hold. In hyperbolic geometry, through a point not on a given line there are infinitely many parallels; in elliptic (spherical) geometry, there are none. These geometries model curved surfaces and spaces.

Trigonometry in Geometry

The use of sine, cosine, tangent, and their inverses to relate angles and side lengths in triangles. The law of sines and the law of cosines generalize right-triangle trigonometry to all triangles, enabling solutions of oblique triangles.

Geometric Proofs

Logical arguments that establish the truth of geometric statements using axioms, definitions, and previously proven theorems. Proof techniques include direct proof, proof by contradiction, and coordinate proof, all of which develop rigorous mathematical reasoning.

Key Terms at a Glance

Angle:A figure formed by two rays sharing a common endpoint (vertex), measured in degrees or radians.

Arc:A continuous portion of the circumference of a circle, measured by its central angle in degrees or by its length.

Bisector:A line, ray, or segment that divides an angle or a segment into two equal parts.

Chord:A straight line segment whose endpoints both lie on a circle. The longest chord of a circle is the diameter.

Circumference:The total distance around a circle, calculated as $C = 2\pi r$ or $C = \pi d$, where $r$ is the radius and $d$ is the diameter.

Congruent:Figures that have exactly the same shape and size. Congruent figures can be mapped onto each other using rigid transformations.

Cosine:In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.

Diameter:A chord that passes through the center of a circle; its length is twice the radius.

Dilation:A transformation that changes the size of a figure by a scale factor relative to a fixed center point, preserving shape but not necessarily size.

Hypotenuse:The side opposite the right angle in a right triangle; it is always the longest side of the triangle.

Isometry:A transformation that preserves distances and angle measures. The four isometries are translation, rotation, reflection, and glide reflection.

Median:A segment connecting a vertex of a triangle to the midpoint of the opposite side. The three medians intersect at the centroid.

Midpoint:The point that divides a line segment into two equal parts, located at the average of the endpoints' coordinates.

Parallel Lines:Lines in the same plane that do not intersect, no matter how far extended. In coordinate geometry they have equal slopes.

Perimeter:The total distance around the boundary of a two-dimensional figure, calculated by summing all side lengths.

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