Geometry Basics

This deck walks you through the foundational vocabulary of geometry, starting from the simplest building blocks like points, lines, rays, and line segments, and working up to broader ideas such as planes and the relationships between figures. You'll also explore angle types, from acute and obtuse to right, straight, and vertical angles, along with the special pairings of complementary and supplementary angles. It's a friendly entry point into how geometers describe shapes, space, and the way objects sit in relation to one another.

It's a great fit if you're just beginning a geometry course, reviewing basics before a test, or strengthening your mental model after a long break from math. Anyone who wants to feel more confident reading geometry problems and diagrams will find these terms useful to internalize, since nearly every later topic depends on this shared vocabulary.

Because these definitions are short and abstract, spaced repetition tends to work really well: come back to the cards frequently over a few days rather than trying to memorize them in one sitting. When you study each term, try to picture or sketch an example, since geometry is deeply visual and pairing a mental image with a definition makes the idea stick much faster.

Geometry begins with a handful of fundamental ideas that describe the space around us. A point is the simplest of these: it represents a single location in space but has no size, no width, no length, and no depth. Because points have no dimensions, we represent them visually with dots and label them with capital letters such as A or B. A line is the next step up; it is a straight, one-dimensional figure that extends infinitely in both directions with no thickness. Any two distinct points define a unique line, which is why two points are always collinear, meaning they lie on the same straight path.

From the idea of a line we get two related figures. A ray is a part of a line that has one fixed endpoint and extends infinitely in just one direction, like a beam of light starting at a flashlight. A line segment, by contrast, is bounded by two distinct endpoints and has a definite length that can be measured. While lines and rays go on forever, segments are finite pieces. A plane, the two-dimensional counterpart of a line, is a flat surface that extends infinitely in every direction. Three non-collinear points are always enough to define a unique plane, and any three points are automatically coplanar.

These objects also help us describe collections of points and shapes. Points are collinear if they all lie on the same straight line, and points are coplanar if they all lie on the same plane. Four or more points may or may not be coplanar, depending on whether they share a flat surface. Finally, shapes themselves can be classified by how they bend inward. A convex shape has all interior angles less than 180°, and any line segment drawn between two interior points stays entirely inside the shape. A concave shape has at least one interior angle greater than 180°, creating an indentation where a connecting segment would pass outside the boundary.

An angle is formed wherever two rays share a common endpoint, called the vertex, and angles are measured in degrees based on the amount of rotation between them. The basic types describe where the angle falls on the 0° to 180° scale. An acute angle measures greater than 0° and less than 90°, with familiar examples including 30°, 45°, and 60°. An obtuse angle measures greater than 90° and less than 180°, such as 120°. A right angle measures exactly 90° and is often marked with a small square at the vertex to distinguish it from other angles. A straight angle measures exactly 180° and forms a flat line, essentially a half-turn.

Angles can also be grouped by how they relate to one another. Complementary angles are two angles whose measures add up to exactly 90°, so a 35° angle and a 55° angle are complementary. Supplementary angles are two angles whose measures add up to exactly 180°, so 110° and 70° form a supplementary pair. When two lines cross, the pairs of opposite angles they create are called vertical angles, and vertical angles are always congruent, meaning they have the same measure. Adjacent angles share a common vertex and a common side but do not overlap; they simply sit next to each other.

Understanding these angle categories is essential because they appear constantly in proofs and calculations. The fact that vertical angles are equal and that adjacent angles on a straight line sum to 180° lets us find unknown angle measures whenever a figure has intersecting or straight lines. Combined with the right angle and the straight angle, these building blocks provide the vocabulary needed to describe everything from simple triangles to the more complex relationships that arise when parallel lines meet a transversal.

Triangles are among the most studied shapes in geometry, and they can be sorted in two main ways. Classified by side length, an equilateral triangle has all three sides equal, an isosceles triangle has exactly two sides equal, and a scalene triangle has no equal sides. Each side classification carries angle consequences as well. Classified by angle, an acute triangle has all three angles less than 90°, a right triangle has exactly one 90° angle, and an obtuse triangle has one angle greater than 90°. Every triangle, regardless of type, obeys the Triangle Angle Sum Theorem: the three interior angles always add up to exactly 180°. This single fact is the key to finding any unknown angle in a triangle when the other two are known.

Beyond classification, triangles can be compared with one another through congruence and similarity. Two triangles are congruent if they have exactly the same size and shape, and there are four standard ways to prove this. The Side-Side-Side (SSS) postulate says that if all three pairs of sides are equal, the triangles are congruent. The Side-Angle-Side (SAS) postulate requires two sides and the angle between them to match. The Angle-Side-Angle (ASA) postulate requires two angles and the side between them to match. The Angle-Angle-Side (AAS) theorem requires two angles and a non-included side. Similar triangles, on the other hand, have the same shape but may differ in size: their corresponding angles are equal and their corresponding sides are in proportion. By the Angle-Angle (AA) similarity postulate, matching two pairs of angles is enough to guarantee similarity.

Two more important theorems govern triangle behavior. The Triangle Inequality Theorem states that the sum of any two side lengths must be greater than the third side, and this must hold for all three combinations. If a, b, and c are the sides, then a + b > c, a + c > b, and b + c > a; otherwise, the three lengths cannot form a triangle. The Exterior Angle Theorem describes the relationship between an exterior angle and the two non-adjacent interior angles (called remote interior angles): the exterior angle equals the sum of those two remote interior angles. Together, these tools let us determine whether triangles can exist, find missing measurements, and compare triangles precisely.

The most famous result in all of geometry applies specifically to right triangles. The Pythagorean Theorem states that in any right triangle with legs of length a and b and hypotenuse of length c, the equation a² + b² = c² holds true. The hypotenuse is the side opposite the right angle, and it is always the longest side of the triangle. A classic example uses legs of length 3 and 4: 3² + 4² = 9 + 16 = 25, so c = √25 = 5. The triple (3, 4, 5) is the most well-known Pythagorean triple, but others include (5, 12, 13) and (8, 15, 17).

The Pythagorean Theorem also has a useful converse. If the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle. Even when the relationship does not hold exactly, we can use it to classify a triangle. If c² < a² + b², where c is the longest side, the triangle is acute. If c² > a² + b², the triangle is obtuse. This makes the Pythagorean relationship a powerful diagnostic tool for any triangle.

Two special right triangles appear so often that their side ratios are worth memorizing. In a 45-45-90 triangle, the two legs are equal and the sides are in the ratio 1 : 1 : √2. So if each leg has length x, the hypotenuse is x√2. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. The side opposite 30° is the shortest, the side opposite 60° is √3 times that length, and the hypotenuse is twice the shortest side. Knowing these shortcuts saves considerable time when solving problems involving right triangles.

A circle is the set of all points in a plane that are the same distance, called the radius, from a central point. The distance across the circle through the center is called the diameter and equals twice the radius. The circumference, or distance around the circle, is given by C = 2πr or equivalently C = πd. The area enclosed by the circle is A = πr², which can also be written as A = π(d/2)² using the diameter. These two formulas are the foundation for nearly every other circle calculation.

Several important line and segment types interact with circles. A chord is a line segment whose two endpoints both lie on the circle, and the longest possible chord in any circle is the diameter. A tangent is a line that touches the circle at exactly one point, called the point of tangency, and a tangent is always perpendicular to the radius drawn to that point. A secant is a line that intersects the circle at exactly two points and, unlike a chord, extends infinitely in both directions beyond the circle.

Arcs and angles have rich relationships inside a circle. A central angle has its vertex at the center of the circle and its sides are radii; the measure of a central angle equals the measure of its intercepted arc. An inscribed angle, by contrast, has its vertex on the circle and its sides are chords, and it always measures half of the central angle that subtends the same arc. This also means that an inscribed angle is exactly half the measure of its intercepted arc. The arc length along a curved section is s = (θ/360°) × 2πr when θ is measured in degrees, or s = rθ when θ is in radians. The area of a sector (the pie-slice region bounded by two radii and an arc) is A = (θ/360°) × πr², which represents the same fraction of the whole circle as the central angle does of 360°.

A regular polygon is a polygon with all sides equal in length and all interior angles equal in measure. Common examples include equilateral triangles, squares, and regular hexagons. The sum of the interior angles of any polygon with n sides is given by (n − 2) × 180°, so a hexagon with 6 sides has interior angles summing to 720°. For a regular n-sided polygon, each individual interior angle is (n − 2) × 180° / n; a regular pentagon, for example, has interior angles of 108° each. The sum of the exterior angles of any convex polygon is always 360°, regardless of how many sides it has. A polygon with n sides has n(n − 3) / 2 diagonals, so a hexagon has 9 diagonals.

For regular polygons, a useful measurement is the apothem, the perpendicular distance from the center of the polygon to the midpoint of any side. The area of a regular polygon is A = ½ × p × a, where p is the perimeter and a is the apothem. The perimeter itself is simply P = n × s, where s is the length of one side and n is the number of sides. These formulas turn regular polygons into straightforward calculations.

Other common two-dimensional shapes have their own area formulas. A rectangle has area A = length × width and perimeter P = 2(length + width). A triangle has area A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex. A parallelogram has area A = base × height, with the height being the perpendicular distance between the two parallel bases, and perimeter P = 2(a + b). A trapezoid has area A = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the lengths of the two parallel bases. A rhombus has area A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of its diagonals. When only the three side lengths of a triangle are known, Heron's formula gives the area directly: A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2 is the semi-perimeter.

Three-dimensional solids extend these ideas by adding height. A rectangular prism has volume V = length × width × height and surface area SA = 2(lw + lh + wh). A cylinder has volume V = πr²h and total surface area SA = 2πr² + 2πrh, where the first term represents the two circular bases and the second the curved lateral surface. A cone has volume V = ⅓πr²h, exactly one-third the volume of a cylinder with the same base and height, and total surface area SA = πr² + πrl, where l is the slant height given by l = √(r² + h²). A sphere has volume V = (4/3)πr³ and surface area SA = 4πr², which is four times the area of a great circle. A pyramid has volume V = ⅓ × base area × height, so a square pyramid with base side s has volume V = ⅓s²h. A triangular prism has volume V = base area × length, which simplifies to V = ½bh × l for a triangle with base b, height h, and prism length l. A hemisphere, being half a sphere, has volume V = (2/3)πr³ and total surface area SA = 3πr², combining the curved portion (2πr²) with the flat circular base (πr²).

Coordinate geometry connects algebra and geometry by placing points on a grid. The distance between two points (x₁, y₁) and (x₂, y₂) is given by d = √[(x₂ − x₁)² + (y₂ − y₁)²], a formula derived directly from the Pythagorean Theorem. The midpoint of a segment with these endpoints is M = ((x₁ + x₂)/2, (y₁ + y₂)/2), which gives the point exactly halfway between them. The slope of a line through two points is m = (y₂ − y₁)/(x₂ − x₁), and slope measures both the steepness and the direction of a line. Lines can be written in several useful forms. The slope-intercept form y = mx + b makes the slope m and y-intercept b immediately visible. The point-slope form y − y₁ = m(x − x₁) is convenient when you know a slope and one point. The standard form Ax + By = C, where A, B, and C are integers with A non-negative, makes it easy to read off intercepts. To find the equation of a line through two points, first calculate the slope using the slope formula, then substitute m and one of the points into point-slope form and simplify.

Several other formulas rely on coordinates. The area of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) is A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|, known as the Shoelace formula for three points. A circle with center (h, k) and radius r has the standard-form equation (x − h)² + (y − k)² = r². The shortest distance from a point (x₀, y₀) to a line Ax + By + C = 0 is d = |Ax₀ + By₀ + C| / √(A² + B²), which always measures the perpendicular distance.

Transformations move figures around the coordinate plane. A translation slides every point the same distance in the same direction, mapping (x, y) to (x + a, y + b) and preserving size, shape, and orientation. A reflection flips a figure over a line of reflection, creating a mirror image that preserves size and shape but reverses orientation. Reflecting (x, y) over the x-axis gives (x, −y); reflecting over the y-axis gives (−x, y). A rotation turns a figure around a fixed center by a specified angle, preserving size and shape. Rotating (x, y) by 90° counterclockwise about the origin gives (−y, x), while a 90° clockwise rotation gives (y, −x). A dilation enlarges or reduces a figure by a scale factor k from a center point, mapping (x, y) to (kx, ky) when the center is the origin.

When two lines are in the same plane and never intersect, no matter how far they are extended, they are parallel and share the same slope. Lines that meet at a 90° angle are perpendicular, and the product of their slopes is −1; if one slope is m, the perpendicular slope is −1/m. A transversal cuts across two lines and creates special angle pairs: corresponding angles (same side of transversal, same position relative to the lines) are congruent when the lines are parallel; alternate interior angles (opposite sides of transversal, between the lines) are congruent; alternate exterior angles (opposite sides of transversal, outside the lines) are congruent; and co-interior or same-side interior angles (same side of transversal, between the lines) are supplementary, summing to 180°. Any one of these angle relationships can be used in reverse to prove that two lines are parallel.

Trigonometry extends right-triangle work with ratios of side lengths. For an angle θ in a right triangle, the sine is sin θ = opposite / hypotenuse, the cosine is cos θ = adjacent / hypotenuse, and the tangent is tan θ = opposite / adjacent, which also equals sin θ / cos θ. The mnemonic SOH-CAH-TOA captures these: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. The reciprocal functions extend these ideas: cosecant csc θ = 1/sin θ, secant sec θ = 1/cos θ, and cotangent cot θ = 1/tan θ. A fundamental identity, sin²θ + cos²θ = 1, follows from the Pythagorean Theorem applied to the unit circle. To find an unknown side, identify the angle and which sides are involved, choose the matching ratio, set up the equation, and solve. For instance, knowing angle θ and the hypotenuse gives opposite = hypotenuse × sin θ, while knowing an angle and the opposite side gives hypotenuse = opposite / sin θ. Together, coordinate geometry, transformations, and trigonometry provide a powerful toolkit for analyzing geometric figures from multiple perspectives.