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).

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.

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 = |
Substitute (2) in (1) and transpose. |

- Q.E.D

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.

- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)

- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more

- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)

- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)

- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle

- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved