Criteria for Congruence of Triangles

In the figure $ABC$ is an equilateral triangle with side $14$ cm. $AX=\frac{1}{3}AB, BY=\frac{1}{3}BC \text{and} CZ=\frac{1}{3}AC$. What is the area (in ${\mathrm{cm}}^2$ ) of $△PQR$?

In triangle ABC as shown in the figure.If AD, BE, CF congruent at O 

 Mark true or false- 

    1.      $\frac{BD}{DC}=\frac{arBDO}{arDCO}$

    2.      $\frac{BD}{DC}=\frac{arABD}{arADC}$

   3.      $\frac{BD}{DC}=\frac{arABO}{arACO}$

If $AD$ is an altitude of an isosceles triangle $ABC$ in which $AB = AC$. Then:

If $ABC$ and $DBC$ are two isosceles triangles on the same base $BC$. Then:

$ABC$ is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively. Then:

If $E$ and $F$ are the midpoints of equal sides $AB$ and $AC$ of a triangle $ABC$. Then:

Which of the following is not a criterion for congruence of triangles?

In triangles $ABC$ and $DEF$, $AB=FD$  and $\angle A=\angle D$ . The two triangles will be congruent by $\mathrm{SAS}$ axiom if :

The correct criterion for $△\mathrm{BGE}\cong △\mathrm{BGF}$ is