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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.

* Negative angles do occur in trigonometry however.

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 step-by-step instructions

The above animation is available as a printable step-by-step 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.
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