

Difference of two angles
Geometry construction using a compass and straightedge
This construction shows how to create an angle which is the difference between two given angles.
This is very similar to Sum of angles,
except that the second angle is drawn inside the first, effectively subtracting the angles.
The two types of construction (add , subtract) can be combined in any way you like. For example, for the four angles P,Q,R,S
you could construct P+Q(R+S).
What if the second angle is larger than the first?
If the second angle is larger than the first then the points M (and so S) will be below the line PQ. The difference angle will still be
∠SPQ as before.
If for example you 'subtracted' 90° angle from a 30°angle, arithmetically this would be negative:
But in geometry, you cannot have negative angles* so the difference between the two angles is 60°.
So the precise definition of what this construction does is, for two angles A and B, it finds the absolute value of a–b
The two vertical bars mean "absolute value" which is always positive, regardless of the way a–b comes out.
This is why the construction is titled "Difference of two angles" – it is not quite the same as arithmetic subtraction.
Proof
This is the same drawing as the last step in the above animation.

Argument 
Reason 
1 
m∠RPS + m∠SPQ = m∠RPQ 
Adjacent angles 
2 
m∠BAC = m∠RPS 
Copied using the procedure in Copying an angle. See that page for proof. 
3 
m∠SPQ =  m∠RPQ  m∠BAC  
Substitute (2) in (1) and transpose. 
 Q.E.D
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Try it yourself
Click here for a printable worksheet containing two angle segment copying problems.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
(C) 2011 Copyright Math Open Reference. All rights reserved

