Exercise 19 Page 647

We are asked to decide if the following statement is always, sometimes, or never true.

First, we can think of an example to the statement. This pair of prisms needs to satisfy two conditions.

1. Have congruent bases.
2. Be similar. By definition, two prisms are similar if their linear measurements are proportional.

There is only one way that these two conditions are met at the same time. The prisms need to be exactly alike. We know this because congruent bases are similar with a scale factor of $1.$ As a result, the other sides also need to be congruent.

Now, let's try to think of a counterexample. This means finding a pair of prisms with congruent bases that are not similar. Notice that there are an infinite number of such pairs of prisms.

These two prisms have congruent bases. Their height are different though. Since the lengths of the two prisms are not proportional, the prisms cannot be similar. Let's now take a look at the statement one more time.

The statement is sometimes true since we found both an example and a counterexample to it.