If all the sides of $△$CAT are equal to corresponding sides of $△$RAT, then $△$RAT will be congruent to $△$CAT and the congruence criterion will be SSS.

$\text{Consider}\Delta ABC\text{and}\Delta ADC,AC=AC\text{(Common side of both triangles)}AB=AD\text{(Sides of isosceles triangle)}BC=CD\text{(Median divides a side in two equal parts)}\therefore \Delta ABC\cong \Delta ADC\left(\text{by SSS congruence criterion}\right)\text{So the option}"\Delta ABC\cong \Delta ADC"\text{is correct.}\therefore \angle ACB=\angle ACD\left(CPCT\right)\text{But they are supplementary angles. So,}\angle ACB+\angle ACD={180}^{\circ }\Rightarrow \angle ACB=\angle ACD={90}^{\circ }\therefore AC\text{is perpendicular to}BD\text{and the hence option "AC is perpendicular to BD" is correct.}\text{Also,}\angle BAC=\angle DAC\text{(corresponding angles of congruent triangles)}\text{Hence, the option}"\angle BAC=\angle DAC"\text{is also correct.}\text{Also, area of}\Delta ABC=\text{area of}\Delta ADC=\frac{1}{2}\left(\text{area of}\Delta ABD\right).\Delta ABC\cong \Delta ADC⟹ar\left(\Delta ABC\right)=ar\left(\Delta ADC\right)\text{Hence, the option "Area of}\Delta ABD=\frac{1}{2}\times \text{Area of}\Delta ABC"\text{is also correct.}$

In $△$ACD and $△$ABD
$\angle BAD=\angle CAD$
[$\because$ AD is the bisector of $\angle A$]
AD = AD        [common side]
AB = AC         [Given]
$△ACD\cong △ABD$ [SAS congruency]
The triangles are congruent by SAS congruence rule.

In $△ACB$ and $△CAD$,
$AD=BC$                [Given]
$\angle CAD=\angle ACB$    [Alternate angles]
$CA=AC$                [common side]
$△ACB\cong △CAD$   [SAS congruency]
$\therefore AB=DC$             [CPCT]

In the $△ABC$ and $△ABD$,
AC = AD   (Given)
CB = DB   (Given)
AB is the common side.
$⟹△ABC\cong △ABD$ [SSS congruency]
Thus, by SSS congruency rule, the two triangles (ABC and ABD) are congruent.

The given statement is false. Even when two triangles have all angles same, they can still have sides of different lengths. However, the ratio of lengths of corresponding sides will be same. But in this case, they will be called 'similar' triangles, not congruent.
Congruent triangles are a special case of similar triangles.
For example, in the image below, both triangles are equilateral and have all angles equal but they are not congruent.

$\angle A+\angle B+\angle C={180}^{\circ }$ (angle sum property of triangle)
$\Rightarrow \angle A+{45}^{\circ }+{45}^{\circ }={180}^{\circ }\Rightarrow \angle A={90}^{\circ }$
The side opposite to largest angle will be the longest side. Hence BC is the largest side.

$\text{In the given triangle,}AB=AD\text{(Given)}AC\text{is the common side}\angle ACB=\angle ACD={90}^{\circ }$
$\therefore \Delta ACB\cong \Delta ACD$  [ RHS criteria]
Hence, by RHS congruence rule, the two triangles are congruent.