In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x$,$MB = x-2$,$AN = x+2$ and $NC = x-1$,find the value of $x$.

In $\Delta ABC$,$A-D-B$,$A-E-C$ and $BD = CE$. If $m\angle B = m\angle C$,prove that $\overline{DE} \parallel \overline{BC}$.

Is the triangle with sides $25 \, cm$,$5 \, cm$,and $24 \, cm$ a right triangle? Give reasons for your answer.

In $\Delta ABC$ and $\Delta PQR$,$\angle A \cong \angle P$ and $\angle B \cong \angle R$. Then,the correspondence $ABC \leftrightarrow \ldots$ is a similarity.

Prove that the sum of the squares of the sides of a rectangle is equal to the sum of the squares of its diagonals.

In $\Delta ABC$ and $\Delta DEF$,$\frac{AB}{DF} = \frac{BC}{EF}$ and $\angle B \cong \angle F$. Then,by $\ldots \ldots \ldots$ condition,the correspondence $ABC \leftrightarrow DFE$ is a similarity.