Shop Essentials (Applied Mathematics)

Geometry: Lines and Angles 151

The class Geometry: Lines and Angles discusses the basic building blocks of all geometry: the line and the angle. Every print used in manufacturing is composed of lines and angles which must be interpreted to manufacture the depicted part. Though part geometry can be incredibly complex, all geometric prints can be broken down into simpler lines and angles. The relationships between the various angles formed when lines intersect can be used to solve geometry problems and interpret blueprints. An understanding of lines and angles is fundamental to learning and applying geometry as well as trigonometry and calculus. After taking this class, users should have a grasp on the types of lines and angles used in geometry, the angles that are formed by intersecting lines, and tranversals. An understanding of the basics of geometry is necessary in various fields including inspection, part program applications, and other important areas of manufacturing.

  • Difficulty Beginner

  • Format Online

  • Number of Lessons 16

  • Language English


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Course Outline
  • The Basics of Geometry
  • Points, Lines, and Rays
  • Angles
  • Types of Angles
  • Angle Pairs
  • Angles and Angle Pairs Review
  • Perpendicular and Parallel Lines
  • Variables
  • Intersecting Transversals
  • Intersecting Transversal: Sample Problem
  • Lines, Variables, and Intersecting Transversals Review
  • Drilling Holes with Slots
  • Drilling Holes with Slots: Sample Problem
  • Bolt Circles
  • Bolt Circles: Sample Problem
  • Drilling Holes with Slots and Bolt Circles Review
  • Define geometry.
  • Define points, lines, and rays.
  • Identify the parts of an angle.
  • Describe the different types of angles.
  • Define the different types of angle pairs.
  • Define perpendicular, parallel, and plane.
  • Describe variables.
  • Describe the angles created by transversals.
  • Solve for unknown variables on a transversal.
  • Solve for unknown variables using angle pairs.
  • Solve for unknown variables using angle pairs.
  • Describe how to solve a bolt circle problem with angle relationships.
  • Solve a bolt circle problem using angle relationships.
Vocabulary Term

acute angle

An angle that measures less than 90 degrees. Two acute angles are present in a right triangle.

adjacent angles

Two angles next to each other that share a common side and common vertex. Adjacent angles form a larger single angle.

alternate angles

Two angles that are located on opposite sides of the transversal line. Alternate angles will equal each other if the lines intersected by the transversal are parallel.

alternate interior angles

Angles created by an intersecting transversal within two parallel lines that are equal to one another. Alternate interior angles are types of vertical angles.


The combination of either two rays with the same endpoint or two straight lines that intersect. Angles are measured in degrees, which specify the amount of separation between the sides of the angle.


The combination of either two rays with the same endpoint or two straight lines that intersect. Angles are measured in degrees, which specify the amount of separation between the sides of the angle.

complementary angles

Two angles that together measure exactly 90 degrees. For example, angles measuring 52° and 38° are complementary angles.

Computer-aided design

CAD. A computer system used to design a model of a product. Geometry is required for CAD.

corresponding angles

Angles created by a transversal that crosses other lines. Corresponding angles have the same measurement when a transversal crosses lines that are parallel.


A common unit of measurement used to measure the size of an angle. Degrees are represented by a small circle positioned above and to the right of a number.

exterior angles

An angle located outside two lines or outside a closed figure. Exterior angles can be equal to each other, depending on the relationship between the lines.


The branch of mathematics that involves the measurements, properties, and relationships of all shapes and sizes of things. Geometry combines simple shapes such as circles, triangles, and squares to create more complex shapes.

interior angles

An angle located between two lines or within a closed figure. Interior angles can can be equal to each other or supplementary.


To meet, cut across, or overlap. Two intersecting lines form the shape of an "X."


A series of points that extends endlessly in two directions. A line is perfectly straight, and its length cannot be measured.

line segment

A section of a line with two endpoints that indicate where the section ends. The length of a line segment can be measured.


A series of points that extends endlessly in two directions. A line is perfectly straight, and its length cannot be measured.


The study of numbers and quantities and their relationships. Mathematics requires an understanding of the logic and rules used to solve numerical problems.

obtuse angle

An angle that measures greater than 90 degrees but less than 180 degrees. An obtuse angle cannot be present in a right triangle.

parallel lines

Two lines in the same plane that, no matter how far they extend, do not intersect with each other. Parallel lines are the same distance apart at any given point.

perpendicular lines

Two lines that intersect to form a 90° angle. The measurement of the angle formed by perpendicular lines may be indicated with a square instead of a number of degrees.


A flat surface that extends infinitely in any direction in three dimensions. A plane is represented on blueprints by a square or rectangle.


A dot that indicates a definite position or location. A point has no width, depth, or length.


The creation of program codes and instructions used to run a machine tool controlled by a computer. CNC programming requires the use of geometry.


A portion of a line that has only one endpoint and extends infinitely in one direction. The length of a ray cannot be measured.

reflex angle

An angle that measures greater than 180 degrees but less than 360 degrees. Reflex angles may help determine the value of unknown angles.

right angle

An angle that measures exactly 90 degrees. When a right angle is present in a triangle, the remaining angles must be acute.

straight angle

An angle that measures exactly 180 degrees. The sides of many geometric forms are straight angles.

supplementary angles

Two angles that combine to measure exactly 180 degrees. Supplementary angles are also adjacent angles.

supplementary interior angles

Two angles whose measurements equal 180 degrees. Supplementary angles appear as two interior angles on the same side of the transversal.


A line that intersects two or more lines at different points. A transversal forms several different angles with the line that it intersects.


Symbols, such as a letters of the alphabet, that represent an unknown quantity. X and Y are commonly used as variables.


The point of an angle where its two sides meet. On some prints, an angle may be identified only by its vertex.

vertical angles

Two angles opposite each other that share the same vertex and are formed by intersecting lines. Vertical angles always have the same measurement in degrees.