State the correspondence between the parts, if 

$△\mathrm{PQR}\cong △\mathrm{XYZ}$

State the correspondence between the parts, if 

$△\mathrm{OPT}\cong △\mathrm{FUG}$

In the adjoining figure, $\mathrm{DC}\parallel \mathrm{AB}$, $\mathrm{AD}=\mathrm{BC}$. Prove using the concept of congruent triangles that $\mathrm{AE}=\mathrm{BF}$.

Draw a diagram for the following statement and prove the result.

In an isosceles triangle, the median joining the vertex(formed by intersection of equal sides) to the midpoint of opposite sides is also an attitude.

Draw a diagram for the following statement and prove the result.

Any two altitudes of an equilateral triangle are congruent.

In the adjoining figure, $DA=CB$ and $AX=BY$. Prove that $△DAY\cong △CBX$

In $△\mathrm{ABC}$, $\angle \mathrm{A}=\angle \mathrm{C}$ and $\mathrm{BM}$ bisects $\angle \mathrm{ABC}$. Prove that $\mathrm{BM}$ is perpendicular to $\mathrm{AC}$.

In the adjoining figure, $\mathrm{O}$ is the midpoint of $\mathrm{PQ}$ and $\angle \mathrm{A}=\angle \mathrm{B}$. Prove that $△\mathrm{PAO}\cong △\mathrm{QBO}$.