# Angles and their relationships

Geometry establishes a lot of imaginary objects and relationships between them in order to define models and the real world.  Angles are an important set of those relationships.  But, we often skip or forget types of relationships between angles.  Let’s look at related angles.  Related angles are pairs of angles that have some sort of relationship to each other.  Several types of related angles are established by Geometry.  Some may surprise, as they aren’t commonly known.

### Types of related angles

Complementary angles – a pair of angles with a common vertex and a sum of a right angle (90°).

Supplementary angles – a pair of angles with a common vertex and a sum of a straight angle (180°).

Explementary angles – a pair of angles with a common vertex and a sum of a full circle (360°).

Vertically opposite angles – a pair of angles that equal to each other and are vertical-and-opposite of each other with a common vertex.  These angles are formed by two intersecting lines.

Of course, a single complementary angle is one of the pair of complementary angles.  A single supplementary angle is one of the pair of supplementary angles.  A single explementary angle is one of a pair of explementary angles.  And, a single vertically opposite angle is one of a pair of vertically opposite angles.

### Conjugate?

The term conjugate angles is sometimes used as a synonym for explementary angles.  Technically, conjugate angles is a set of angles with a sum of 360°. Despite the word conjugate meaning coupled/related/connected, it seems that the term conjugate angles is a set that need not be made up of only two angles, and so the angles within the set are not necessarily related angles, though they are connected by a common vertex. Additionally, the term conjugate angles does not apply directly to any angles within the set, but only to the set itself, so there’s no singular form of this term.

### SOLIDWORKS support for angle dimensions

Though explementary and vertically opposite angles are not as common as supplementary and complementary angles, they are important from time to time when designing and defining mechanical components and assemblies.  As such, SOLIDWORKS has supported both explementary and vertically opposite angles since release 2015.  See Year of the Angle Dimension – Part 2 – Flipping out (and over) and Flipped Angle Dimension in SOLIDWORKS for information on how to use these types of angles in your dimension scheme.

### Other types of angle dimensions in SOLIDWORKS

Another type of angle supported in SOLIDWORKS since release 2015 is the straight angle (180°).  You can dimension two lines that form a straight angle.

Also, angle dimensions can be created from one line and one vertex instead of always from two lines.

## Author: fcsuper

As a drafter, mechanical designer and CAD engineer, I've been in the mechanical design field since 1991. For the first 8 years of my career, I was an AutoCAD professional. I utilized AutoLISP and many other AutoCAD customization features to streamline drafting activities for 6+ drafters and designers. I authored several custom functions, one of which was published in the March 1997 issue of Cadalyst Magazine. Since 1998, I've been used SolidWorks non-stop. I've worked to utilize the SolidWorks' user environment to simplify drafting and design activities for 20+ engineers. I've created this website to provide current information about SolidWorks from a variety of contributors. More recently, I am now employed by Dassault Systemes as SOLIDWORKS Sr. Product Definition Manager to improve drawing, annotation and MBD related areas.