Given $R=\{(P_1,P_2):$ $P_1$ and$\; P_2$ have same number of sides$\}$:

For any Polygon w/ $P$ number of sides, $(P,P)\in R$ as same polygon has same number of sides with itself. Hence $R$ is reflexive.

If $P_1$ has same number of sides as $P_2$ the $P_2$ also has same number of sides as $P_1$

Hence it follows that $(P_1,P_2) \in R \Rightarrow (P_2P_1) \in R$ and $R$ is symmetric.

If $P_1$ has same number of sides as $P_2$ and $P_2$ has same number of sides as $P_3$ then $P_1$ and $P_3$ have the same number of slides.

Hence it follows that $R$ is transitive also.

Therefore, $R$ is an equivalence relation since its reflexive, symmetric and transitive.

Consider a rightangle triangle $T$ with the sides $3,4,5$. Since $T$ is a polygon with 3 sides, the the set of all elements in $A$ related to the right angle triangle $T$ is the set of all triangles.