Geometry: Lines and Angles 151

acute angle

An angle that measures less than 90 degrees. Acute angles are used to define sharp corners and narrow features in geometric layouts.

additive manufacturing

AM. A process of creating objects by building them up layer by layer, rather than removing material from a solid block. Additive manufacturing is used to produce complex shapes efficiently.

adjacent angles

Two angles that share a common side and a common vertex. Adjacent angles are used to break larger angles into smaller, measurable sections.

alternate angles

An angle located on the opposite side of a transversal from another angle. Alternate angles are equal if the crossed lines are parallel.

angles

The space formed between two intersecting lines or rays sharing a common vertex. Angles are critical for setting machine orientations and aligning features on a part.

bolt circle

A set of holes arranged around a central point that allows you to attach two circular components using bolts. Bolt circles allow even fastening and balanced load distribution in assemblies such as flanges or pipe fittings.

Cartesian coordinate system

A framework for locating points in space using perpendicular axes that measure position with numerical coordinates. The Cartesian system is used in geometry to describe shapes, analyze motion, and provide a basis for graphical representation.

central angles

An angle formed by two straight lines that meet at the center point of a circle and extend to its circumference. Central angles define spacing around a circle and are used extensively in machining, inspection, and layout work.

complementary angles

Two angles whose measures add up to 90 degrees. Complementary angles are used to determine missing angles in geometric layouts and machine setups.

computer-aided design

CAD. Software used to create precise digital models of parts and assemblies. Computer-aided design allows you to visualize geometry, plan toolpaths, and simulate machining operations.

corresponding angles

One of a pair of angles that occupy the same relative position where a transversal crosses two lines. Corresponding angles are equal when the lines are parallel.

degree measure

A unit used to quantify the size of an angle. Degree measure describes the distance of one side of an angle from the other.

dimensions

A measurable extent of length, width, height, or angle. Dimensions specify the size and position of features on a part to ensure proper fit and function.

endpoints

A fixed point where a line segment or ray stops. Endpoints define limits for cuts, slots, or edges on a part.

exterior angles

An angle formed on the outside of two lines that a transversal intersects. Exterior angles pair with interior angles to reveal geometric relationships.

fixtures

A device used to hold a workpiece securely in place during machining or assembly. Fixtures ensure accurate positioning and repeatable results for multiple parts.

flanges

A projecting rim or edge on a part that allows it to be attached to another component. A flange often contains a bolt circle for secure, leak-free connections.

geometry

A branch of mathematics that studies the size, shape, and spatial relationships of objects. Geometry is used in manufacturing to design, produce, measure, and verify parts for accuracy and assembly.

hole

A round opening that passes partially or completely through a material. A hole can provide clearance for fasteners, allow fluid flow, or create alignment points in a part.

inspection

The process of measuring and evaluating parts to verify compliance with specifications. Inspection identifies errors or deviations before parts are assembled or shipped.

interior angles

An angle formed on the inside between two lines that a transversal intersects. Interior angles are key to solving for unknown measurements when lines are parallel.

intersect

To meet or cross at a single point. Intersecting points define vertices and angles in geometric layouts and part features.

laser

A focused beam of light used for measuring, cutting, or marking parts. Lasers provide precise guidance for machining and layout operations.

line segment

A portion of a line that has two distinct endpoints. Line segments represent finite edges or features on prints and models.

linear axes

A single straight path along which motion occurs in only one direction and its opposite, such as left–right or up–down. Linear axes are used in machines and geometry to constrain movement simply and precisely, forming the building blocks of more complex multi-axis systems.

lines

A straight path that extends in two directions without end. Lines define edges, center axes, and paths of motion for parts and tools.

machining

A manufacturing process that removes material from a workpiece to create a desired shape or surface finish. Machining is commonly used to produce precise parts by cutting, drilling, milling, or turning metal and other solid materials.

mathematics

The study of numbers, quantities, shapes, and their relationships using symbolic representations. Mathematics provides the foundation for calculating measurements, tolerances, and dimensions in manufacturing.

mating

The condition of a part that is designed to fit or connect precisely with another part. Mating surfaces ensure proper alignment, engagement, and function in an assembly.

models

A digital or physical representation of a part or assembly. Models help you visualize features, verify dimensions, and plan manufacturing operations.

obtuse angle

An angle that measures more than 90 degrees but less than 180 degrees. Obtuse angles define wide corners and sloped surfaces in geometry.

one-dimensional

1D. Having only a single measurable extent, such as length, without width or height. One-dimensional objects are used in geometry to represent the simplest forms, like lines or line segments, which serve as the basis for more complex shapes.

parallel

Describes lines or surfaces that are always the same distance apart and never intersect. Parallel geometry helps maintain uniform spacing and consistent part dimensions.

perpendicular

Describes two lines or surfaces that meet to form a 90-degree angle. Perpendicular features are common in prints that specify exact corner or hole placement.

pipe fittings

A component used to connect, redirect, or terminate sections of pipe. Pipe fittings maintain sealed, accurate alignment in plumbing and fluid systems.

planes

A flat two-dimensional surface that extends endlessly in all directions. A plane provides the reference space in which points, lines, and angles exist in geometry.

points

A location in space with no length, width, or depth. Points are used on prints to indicate exact positions for holes, edges, or reference locations.

print

A technical drawing that represents a part’s geometry, dimensions, and features. Prints guide manufacturing, inspection, and assembly processes.

quality control

QC. Procedures and measures used to maintain the desired level of product consistency and performance. Quality control ensures manufactured parts meet design standards and reduce defects.

rays

A straight path that starts at a specific point and extends infinitely in one direction. Rays are used to define angles and directions in applications such as machining and layout.

reflex angle

An angle that measures more than 180 degrees but less than 360 degrees. Reflex angles describe rotations that extend beyond a straight line and help define curved or circular paths.

right angle

An angle that measures exactly 90 degrees. Right angles form perpendicular intersections and are essential for creating square features.

same-side exterior angles

An exterior angle located outside a pair of parallel lines on one side of a transversal. A same-side exterior angle is supplementary to the exterior angle on the same side of the transversal.

same-side interior angles

An interior angle located between a pair of parallel lines on one side of a transversal. A same-side interior angle is supplementary to the interior angle on the same side of the transversal.

sides

The straight edge that forms part of a geometric shape or angle. Sides determine the boundaries and dimensions of angles and features.

slot

A narrow cut or opening with a defined length and width. Slots often guide movement, provide clearance, or allow fasteners to pass through manufactured parts.

straight angle

An angle that measures exactly 180 degrees. Straight angles form a straight line and are used to represent linear alignment in geometry.

supplementary angle pairs

Two interior angles on the same side of a transversal whose measures add to 180 degrees. Supplementary angle pairs reveal relationships between intersecting lines.

supplementary angles

Two angles that share a common side and common vertex and add up to 180 degrees. Supplementary angles define straight-line relationships and help verify feature alignment.

three-dimensional

3D. Having three measurable extents: length, width, and height. Three-dimensional objects are used in geometry to represent solid forms, such as cubes, spheres, and cylinders.

tolerances

The allowable variation in a part’s dimension or geometry. Tolerances ensure parts fit together correctly and maintain consistent quality.

toolpaths

The programmed route that a cutting tool follows to machine a part. Toolpaths are designed to create precise shapes while minimizing cutting time and material waste.

transversal

A line that crosses two or more other lines at different points. A transversal creates multiple related angles that help determine unknown measurements.

two-dimensional

2D. Having both length and width but no depth. Two-dimensional figures are used in geometry to represent flat shapes that lie on a single plane.

variable

A symbol that represents an unknown numerical value in a mathematical problem. Variables allow you to calculate missing dimensions or angle measures in geometry.

vertex

The point where two lines or rays meet to form an angle. A vertex defines the location of angular measurements and is critical for geometric construction and layout.

vertical angles

Two angles that are opposite each other when two lines intersect. Vertical angles always have equal measures and help calculate unknown angle values.

ways

A precisely machined track or surface that guides the movement of machine components. Ways provide smooth, accurate, and aligned motion between parts, such as the sliding elements of a lathe or milling machine.