Congruence of T...

In figure, if $$AB=AC$$ and $$AP=AQ$$, then by which congruence criterion $$\Delta PBC \cong \Delta QCB$$?

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

If $$\Delta ABC \cong \Delta DEF$$ by SSS congruence rule then :

In the given figure, If OA=OB,OD=OC, then $$\Delta AOD \cong \Delta BOC$$ by congruence rule:

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

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

In the given figure, if $$AB=DC$$, $$\angle ABD= \angle CDB$$, which congruence rule would you apply to prove $$\Delta ABD$$ $$\cong \Delta CDB$$ ?

ABC is an isosceles triangle with AB $$= $$AC and D is a point on BC such that  $$AD \perp BC$$ (Fig. 7.13). To prove that $$\angle BAD = \angle CAD,$$ a student proceeded as follows:

$$\Delta ABD$$ and $$ \Delta ACD,$$
AB $$=$$ AC (Given)
$$\angle B = \angle C$$   (because AB $$=$$ AC)
and $$\angle ADB = \angle ADC$$
Therefore, $$\Delta ABD \cong \Delta ACD (AAS)$$
So, $$\angle  BAD = \angle CAD (CPCT)$$
What is the defect in the above arguments?

Given $$\Delta OAP \cong  \Delta OBP$$ in figure, the criteria by which the triangles are congruent is:

If $$\Delta ABC \cong  \Delta DEF$$ by SSS congruence rule then