## Angles and their relationships

Uncommonly known types of related angles, and their SOLIDWORKS support. Some may surprise!

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.

## Year of the Angle Dimension – Part 2 – Flipping out (and over)

In SOLIDWORKS 2015, there are two methods to change (flip) an angle dimension.

#### Vertically Opposite Angle

You can now flip any placed angle dimension to its vertically opposite angle.   This is useful when you wish to place the entire angle dimension outside of the model edges.

1. Right-click on the angle to bring up the right-click menu.
2. Select Display Options, then Vertically Opposite Angle.

#### Explementary Angle

You can now flip any placed angle dimension to its explementary angle.  Here is a way overly complex video about explementary angles.  Here’s a simpler explanation straight from the dictionary.

1. Right-click on the angle to bring up the right-click menu.
2. Select Display Options, then Explementary Angle.

#### Choose Explementary or Vertically Opposite in dimension preview

When creating an angle dimension with Smart Dimension tool, you can now choose between the explementary angle or the vertically opposite angle during the preview by holding down the ALT key when the mouse is in the vertically opposite region.  In SOLIDWORKS 2014 and prior, you were only offered the explementary angle.