Chapter 7.2, Problem 53HP

To determine

To find: The counterexample for the given statement.

Answer to Problem 53HP

The counter-example is that all rectangles are similar as the hypothesis is true and the conclusion false.

Explanation of Solution

Consider that the two polygons are similar when the corresponding angles are similar and the corresponding side length are proportional.

Consider the following rectangles is shown in Figure 1

  

Figure 1

By the theorem of similar polygon the above two figures are congruent.

Then, compare the angles as,

  $\begin{array}{l}\frac{QR}{WX}=\frac{NQ}{WZ}\\ \frac{9}{4}=\frac{18}{10}\\ \frac{9}{4}\ne \frac{9}{5}\end{array}$

Since, $\frac{QR}{WX}=\frac{NQ}{WZ}$ corresponding side lengths are not proportional.

Thus, the counter-example is that all rectangles are similar as the hypothesis is true and the conclusion false.