Question

Write a two column proof. If ZY=3XY, then ZX=4XY
 

Answer 1

1. ZX=ZY+YX	Segment ZX is made up of two adjacent segments ZY and YX
2. ZY=3XY	Given
3. YX≡XY	Length X→Y is the same as length Y→X, so YX and XY are equivalent
4. ZX=3XY+XY=4XY QED	Rewrite 1 substituting for ZY and using the equivalence of XY and YX

Comments

At each step we are dealing with the property length. XY, for example, is a segment with a particular unique length. The segment doesn’t even have to be a straight line, but it will always have a fixed length. XY could be the arc of a circle, for instance. So in this proof we are adding lengths together: in geometry we specify the labels at each end of a segment to mean the length of the segment. When we say ZY=3XY we mean the length from a point Z to a point Y (and vice versa: point Y to point Z) is 3 times the length from point X to point Y (and vice versa).