Chapter 7.2, Problem 52HP

To determine

To find: Whether the equivalence relationship is any relationship that satisfies the reflexive symmetric and transitive properties.

Answer to Problem 52HP

The similarity is the equivalence relation, as it has reflexive, symmetric and transitive properties.

Explanation of Solution

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

Consider that the reflexive property state that if $a=a$ , the figure is congruent to itself if the particular figure is similar to itself that is the similarity is reflexive.

Consider that the property of symmetry states that for $a=b$ and $b=a$ both the congruence and equation have symmetric property as symmetry also have the symmetric property for $\frac{AB}{XY}=\frac{BC}{YZ}$ and $\frac{BC}{YZ}=\frac{AB}{XY}$ .

Consider the transitive property states that for $a=b$ and $b=c$ then $a=c$ . Here, the congruence and the equation has transitive property. Hence, the similarity also have the transitive property if $\frac{AB}{XY}=\frac{BC}{YZ}$ . Then,

  $\begin{array}{c}\frac{AB}{EF}=\left(\frac{AB}{XY}\right)\left(\frac{XY}{EF}\right)\\ =\left(\frac{BC}{YZ}\right)\left(\frac{YZ}{FG}\right)\\ =\frac{BC}{FG}\end{array}$

Therefore, the similarity is the equivalence relation, as it has reflexive, symmetric and transitive properties.