Let's begin with an example. We will create a line segment with an unknown length.
Now, let's measure it with a ruler. First, we measure it where the endpoint lines up with zero on the ruler.
We can see on the ruler that the line segment is 4 cm long. Next, let's see what happens when we move the ruler to the left and line up the left end of the segment with an arbitrary non-zero number. For instance, 2.6.
The left end of the line segment is now aligned with 2.6 on the ruler. The right of the segment is lined up with 6.6. Let's consider these as coordinates on the ruler and calculate the length of the segment using the Ruler Postulate. Since these measurements are neither x- nor y-coordinates we call them r_1 and r_2.
The length of the line segment still measures to be 4 cm. We can, therefore, conclude that we do not need to line up an object at zero on a ruler. This is because, when we move the left side of the ruler, we also move the right side of the ruler the same distance. Your cousin is correct.
