# Axiom 2 and 3: If equals are added or subtracted to equals, the wholes are equal

Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of the “same kind.” Addition or elimination of equal parameters to equal quantities results in equal things, for which the relation of equality and the operation of subtraction make sense. In Euclid’s mathematics this relation and this operation apply not only to straight segments and numbers but also to geometrical objects.

### Objectives

• To demonstrate if equals are added to equals, the wholes are equal
• To demonstrate if equals are subtracted to equals, the wholes are equal

15 minutes

### Prerequisites/Instructions, prior preparations, if any

Prior knowledge of points and lines

### Materials/ Resources needed

• Digital : Computer, geogebra application, projector.
• Non digital : Worksheet and pencil
• Geogebra files : “Axiom-2 and 3.ggb

### Process (How to do the activity)

• The file demonstrates the Euclid's second and third axiom.
• The measures of all the lines corresponds to the distance between 0 of x-axis to point F.
• If the distance of point F from 0 is increased or decreased the lengths of all the lines also varies accordingly this can be done by using the slider Equals1.
• With the change in position of slider Equals2 an equal length of 3 is added to all lines.
• It can be observed that the length of all the lines correspondingly increases.
• What do you notice about the lines.
• Again changing the position of slider Equals2 to 0 the measure of 3 units is eliminated.
• Now what do you notice about the lines.
• Record the segment lengths in the worksheet
 Position of Equals2 0 - Point F Length BC Length GH Length EF Length IJ Length of the line + position of Equals 2 . .

Evaluation at the end of the activity

• Can you conclude if equal measures are added to equal lines the resulting lines are equal.