Geometry and Trigonometry Fundamentals

Comprehensive vocabulary flashcards covering fundamental geometric elements, angle classifications, angle pairs, families of lines, polygons, right-triangle trigonometry, reciprocal ratios, angles of elevation and depression, perimeter, area, and solid volume.

Points

Precise locations in space that have no dimension.

Straight Line

A locus of points that extends infinitely in both directions, having no curvature, and maintaining a constant direction.

Curvature of a Straight Line

The degree of bending in a line, which is zero for a straight line because it maintains a constant direction.

Ray (or Half-Line)

The portion of a line lying to one side of a point (end point) on the line.

End Point of a Ray

The single initial point on a line from which a ray or half-line originates and extends infinitely in one direction.

Line Segment

A portion of a straight line that is lying between two end points.

End Points of a Line Segment

The two fixed boundary points between which a straight line segment lies.

Plane

A flat, two-dimensional surface that extends infinitely in all directions and has no thickness.

Thickness of a Plane

The physical depth of a plane, which is zero because a plane is strictly a two-dimensional surface.

Angle

The measure of the "opening" or gap between two intersecting lines or rays, commonly measured in degrees or radians.

Vertex

The common point of intersection where two intersecting lines or rays meet to form an angle.

Angle Notations

Standard notations used to represent an angle, including a Greek letter ($\theta$), a single vertex letter ($\angle A$), or three letters with the vertex centered ($\angle BAC$).

One Full Rotation

A complete angular turn around a point, equivalent to $360^\circ = 2\pi\text{ rads} = 400\text{ grads}$.

Radian ($\text{rad}$)

A unit of angular measurement in which one full rotation equals $2\pi\text{ rads}$.

Gradian ($\text{grad}$)

A unit of angular measurement in which one full rotation equals $400\text{ grads}$.

Degree ($^\circ$)

A unit of angular measurement in which one full rotation equals $360^\circ$.

Acute Angle

An angle whose measure is strictly between $0^\circ$ and $90^\circ$ ($0^\circ < \theta < 90^\circ$).

Right Angle

An angle whose measure is exactly equal to $90^\circ$ ($\theta = 90^\circ$).

Obtuse Angle

An angle whose measure is strictly between $90^\circ$ and $180^\circ$ ($90^\circ < \theta < 180^\circ$).

Straight Angle

An angle whose measure is exactly equal to $180^\circ$ ($\theta = 180^\circ$).

Reflex Angle

An angle whose measure is strictly between $180^\circ$ and $360^\circ$ ($180^\circ < \theta < 360^\circ$).

Complete Angle

An angle whose measure is exactly equal to $360^\circ$ ($\theta = 360^\circ$).

Complementary Angle Pair

A pair of angles whose measures sum to $90^\circ$ ($\alpha + \beta = 90^\circ$).

Supplementary Angle Pair

A pair of angles whose measures sum to $180^\circ$ ($\alpha + \beta = 180^\circ$).

Explementary / Conjugate Angle Pair

A pair of angles whose measures sum to $360^\circ$ ($\alpha + \beta = 360^\circ$).

Family of Lines

A set of lines that are somehow related to each other.

Family of Parallel Lines

A set of related lines that lie in the same plane, have identical direction, and never intersect.

Family of Intersecting Lines

A set of related lines that all cross each other at a single common point.

Transversal

A line that intersects a family of lines.

Triangle

A closed three-sided two-dimensional geometric figure classified by side lengths and internal angles.

Equilateral Triangle

A triangle in which all three sides are equal in length and all interior angles are equal.

Isosceles Triangle

A triangle that has two sides of equal length.

Scalene Triangle

A triangle that has no equal sides.

Acute Triangle

A triangle in which all interior angles are strictly less than $90^\circ$ (all angles $< 90^\circ$).

Right Triangle

A triangle in which exactly one interior angle is equal to $90^\circ$ (one angle $= 90^\circ$).

Obtuse Triangle

A triangle in which one interior angle is strictly greater than $90^\circ$ (one angle $> 90^\circ$).

Quadrilateral

A four-sided polygon enclosed by four straight line segments.

Square

A quadrilateral with four equal sides and four right angles ($90^\circ$).

Rectangle

A quadrilateral with opposite sides of equal length and four right angles ($90^\circ$).

Parallelogram

A quadrilateral with opposite sides parallel and equal in length.

Rhombus

A quadrilateral with all four sides of equal length and opposite sides parallel.

Trapezoid

A quadrilateral having one pair of parallel opposite sides.

Side Adjacent

The leg of a right triangle that forms one side of a specified acute angle, other than the hypotenuse.

Side Opposite

The leg of a right triangle that is located directly across from a specified acute angle.

Hypotenuse

The longest side of a right triangle, situated directly opposite the right angle ($90^\circ$).

Sine ($\sin(\theta)$)

The primary trigonometric ratio defined as $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Cosine ($\cos(\theta)$)

The primary trigonometric ratio defined as $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Tangent ($\tan(\theta)$)

The primary trigonometric ratio defined as $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Reciprocal Trigonometric Ratios

Trigonometric ratios defined by taking the reciprocal of sine, cosine, and tangent.

Cosecant ($\csc(\theta)$)

The reciprocal trigonometric ratio of sine, defined as $\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}$.

Secant ($\sec(\theta)$)

The reciprocal trigonometric ratio of cosine, defined as $\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}$.

Cotangent ($\cot(\theta)$)

The reciprocal trigonometric ratio of tangent, defined as $\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}$.

Inverse Sine ($\sin^{-1}$)

The inverse trigonometric function used to solve for an angle, given by $\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$.

Inverse Cosine ($\cos^{-1}$)

The inverse trigonometric function used to solve for an angle, given by $\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$.

Inverse Tangent ($\tan^{-1}$)

The inverse trigonometric function used to solve for an angle, given by $\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$.

Common Error: Reciprocal vs. Inverse Sine

The incorrect assumption that the reciprocal of sine equals inverse sine: $\frac{1}{\sin(\theta)} \neq \sin^{-1}(\theta)$.

Common Error: Reciprocal vs. Inverse Cosine

The incorrect assumption that the reciprocal of cosine equals inverse cosine: $\frac{1}{\cos(\theta)} \neq \cos^{-1}(\theta)$.

Common Error: Reciprocal vs. Inverse Tangent

The incorrect assumption that the reciprocal of tangent equals inverse tangent: $\frac{1}{\tan(\theta)} \neq \tan^{-1}(\theta)$.

Angle of Elevation

The angle between the horizontal and the line of sight to an object that is above the horizontal.

Angle of Depression

The angle between the horizontal and the line of sight to an object that is below the horizontal.

Line of Sight

The straight line connecting an observer's eye to the object being viewed.

Horizontal Reference Line

The horizontal baseline from which angles of elevation and depression are measured.

Perimeter

The total distance or length around the outside edge of a two-dimensional closed shape.

Perimeter of a Rectangle

The total boundary distance of a rectangle, given by the formula $P = 2b + 2h$, where $b$ is the base and $h$ is the height.

Perimeter of a Triangle

The total boundary distance of a triangle, given by the formula $P = a + b + c$, where $a$, $b$, and $c$ are the lengths of the sides.

Circumference of a Circle

The total boundary length or perimeter of a circle, calculated using $C = 2\pi r = \pi d$.

Radius ($r$)

The straight distance from the center of a circle to any point on its outer boundary.

Diameter ($d$)

The straight distance across a circle through its center, equal to twice the radius ($d = 2r$).

Area

The amount of two-dimensional space enclosed within the boundary of a closed shape.

Area of a Rectangle

The enclosed two-dimensional space of a rectangle, calculated by $A = b \times h$, where $b$ is the base and $h$ is the height.

Area of a Triangle

The enclosed two-dimensional space of a triangle, calculated by $A = \frac{1}{2}(b \times h)$, where $b$ is the base and $h$ is the height.

Area of a Circle (Radius Form)

The enclosed space of a circle expressed in terms of its radius: $A = \pi r^2$.

Area of a Circle (Diameter Form)

The enclosed space of a circle expressed in terms of its diameter: $A = \frac{\pi}{4}d^2$.

Volume

The total amount of space enclosed by a closed surface.

Volume of a Cone or Pyramid

The volumetric formula shared by both a cone and a pyramid: $V = \frac{1}{3} \times A_{\text{base}} \times h$, where $A_{\text{base}}$ is the base area and $h$ is the perpendicular height.