In general, congruent polygons have the same shape. In other words:

• they have the same number of sides,

• all corresponding sides are congruent, and

• all corresponding interior angles are congruent.

With that in mind, let's compare each of the given polygons to $\mathcal{P}.$

• The polygon $\mathcal{P}$ is a triangle (it has $3$ sides). So, the polygons $\mathcal{R},$ $\mathcal{S},$ and $\mathcal{U}$ can't be congruent to the polygon $\mathcal{P}$ because they do not have $3$ sides.

• The polygon $\mathcal{T}$ is not congruent to $\mathcal{P}$ (even though they both have $3$ sides) because $\mathcal{T}$ and $\mathcal{P}$ don't have the same shape. In particular, notice that $\mathcal{T}$ has a right angle, while $\mathcal{P}$ doesn't have a right angle.

On the other hand, $\mathcal{P}$ and $\mathcal{Q}$ have the same shape: They have congruent corresponding sides and congruent corresponding angles. This implies that $\mathcal{P} \cong \mathcal{Q}.$

In general, congruent polygons have the same shape. In other words:

• they have the same number of sides,

• all corresponding sides are congruent, and

• all corresponding interior angles are congruent.

With that in mind, let's compare each of the given polygons to $\mathcal{P}.$

• The polygon $\mathcal{P}$ is an octagon (it has eight sides). So, the polygons $\mathcal{R},$ $\mathcal{T},$ and $\mathcal{Q}$ can't be congruent to the polygon $\mathcal{P}$ because they do not have $8$ sides.

• The polygon $\mathcal{S}$ is not congruent to $\,\mathcal{P}$ (even though they both have $8$ sides) because $\mathcal{S}$ and $\mathcal{P}$ don't have the same shape. In particular, notice that $\mathcal{P}$ has a right angle, while $\mathcal{S}$ doesn't have a right angle.

On the other hand, $\mathcal{P}$ and $\mathcal{U}$ have the same shape: They have congruent corresponding sides and congruent corresponding angles. This implies that $\mathcal{P} \cong \mathcal{U}.$

In general, congruent polygons have the same shape. In other words:

• they have the same number of sides,

• all corresponding sides are congruent, and

• all corresponding interior angles are congruent.

With that in mind, let's compare the given polygons.

• Statement I is false. Notice that the polygons $\mathcal{P}$ and $\mathcal{Q}$ have the same number of sides, but they don't seem to have the same shape. In particular, we notice they don't have corresponding congruent angles ($\mathcal{P}$ does not have right angles, while $\mathcal{Q}$ does).

• Statement II is false. By the same reasoning as above, we can see that the polygons $\mathcal{Q}$ and $\mathcal{R}$ are not congruent.

• Statement III is true. Since the polygons $\mathcal{P}$ and $\mathcal{R}$ have the same number of sides and all corresponding sides and angles are congruent, we conclude that $\mathcal{P}\cong\mathcal{R}.$

Therefore, the correct answer is "III only."

In general, congruent polygons have the same shape. In other words:

• they have the same number of sides,

• all corresponding sides are congruent, and

• all corresponding interior angles are congruent.

With that in mind, let's compare the given polygons.

• Statement I is false. Notice that the polygons $\mathcal{R}$ and $\mathcal{S}$ have the same number of sides, but they don't seem to have the same shape. In particular, we notice they don't have corresponding congruent sides.

• Statement II is false. By the same reasoning as above, we can see that the polygons $\mathcal{S}$ and $\mathcal{T}$ are not congruent.

• Statement III is true. Since the polygons $\mathcal{R}$ and $\mathcal{T}$ have the same number of sides and all corresponding sides and angles are congruent, we conclude that $\mathcal{T}\cong\mathcal{R}.$

Therefore, the correct answer is "III only."

Notice that $\angle A \cong \angle C.$

Let's determine the vertices in the second quadrilateral that correspond to $A.$ First, notice the following:

\begin{align} AB &=LK \\ AD &= KN\\ \angle A &\cong \angle K \end{align}

Therefore,

• vertex $A$ corresponds to vertex $K,$

• consequently, vertex $C$ corresponds to vertex $M.$

As for the other two vertices, notice we have two more pairs of congruent angles:

\begin{align} \angle B &\cong \angle L\\ \angle D &\cong \angle N. \end{align}

Similarly, we have two more pairs of congruent sides:

\begin{align} CD &= MN\\ AD &= KN, \end{align}

Therefore,

• vertex $B$ corresponds to $L,$ and

• vertex $D$ corresponds to $N.$

Therefore, we conclude that a correct congruence statement is

\[ ABCD \cong KLMN.\]

We have three pairs of congruent angles: \begin{align} \angle A &\cong \angle E\\ \angle B &\cong \angle F\\ \angle D &\cong \angle H. \end{align}

Similarly, we have three pairs of congruent sides: \begin{align} BD &=FH\\ AD &= EH\\ AB &= EF. \end{align}

Since all corresponding sides and angles are congruent, the two triangles are congruent. In particular

• vertex $A$ corresponds to vertex $E,$

• vertex $B$ corresponds to vertex $F,$ and

• vertex $D$ corresponds to vertex $H.$

Therefore, we conclude that a correct congruence statement is \[ \triangle ABD \cong \triangle EFH. \]

The statement $ABCD\cong EFGH$ means that $\mathcal P$ and $\mathcal Q$ are congruent. Moreover,

• vertex $A$ corresponds to vertex $E,$

• vertex $B$ corresponds to vertex $F,$

• vertex $C$ corresponds to vertex $G,$

• vertex $D$ corresponds to vertex $H.$

Therefore,

\[ \begin{align*} HE & = DA \\[3pt] y &= 12. \end{align*} \]

Consequently, \[ \begin{align*} GH &= CD \\[5pt] & = \dfrac{3}{4}y \\[5pt] & = \dfrac{3}{4} \cdot 12 \\[5pt] &= 9. \end{align*} \]

The figure shows that $GH=EF$ and $HE=FG.$ Therefore, the perimeter of $\mathcal{Q}$ is

\[ \begin{align*} p_{\mathcal{Q}} &= EF+FG+GH+HE\\[3pt] &= 9+12+9+12\\[3pt] &= 42. \end{align*} \]

The statement $ABCD\cong EFGH$ means that the quadrilaterals are congruent. Moreover,

• vertex $A$ corresponds to vertex $E,$

• vertex $B$ corresponds to vertex $F,$

• vertex $C$ corresponds to vertex $G,$

• vertex $D$ corresponds to vertex $H.$

Therefore,

\begin{align*} GF&=BC\\[3pt] 2x+1 &= 3\\[3pt] 2x &= 2\\[3pt] x&=1. \end{align*}

Also, we have

\begin{align*} m\angle E&=m\angle A\\[3pt] {(5y)}^\circ &= 65^\circ\\[3pt] 5y &= 65\\[3pt] y&=13. \end{align*}

Finally, \[ y-x = 13-1=12. \]