Exercise 32 Page 394

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 $x\text{-}$ nor $y\text{-}$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.