What is the reason for this geometric answer? 
$$\frac{AB}{CD} = \frac{2}{2.6} = 0.77\ldots$$
$$\frac{AC}{AD} = \frac{2}{2.6} = 0.77\ldots$$
$$\frac{BC}{AC} = \frac{2}{2.6} = 0.77\ldots$$
Therefore $\triangle ABC$ and $\triangle ACD$ are similar.
I know from the answer sheet that $y$ is $47^\circ$ and $x$ is $109^\circ$. I don't know the reasons why though. The diagram does not look like a parallelogram because both opposite sides are not parallel. 
 A: The 3 relations you wrote imply that $ABC$ is similar to $ACD$ hence $\hat{ACD}=x$ and $\hat{ADC}=y$ because of the similarity of the 2 triangles.
Also you get that $\hat{BCA}=\hat{CAD}$.
The reason is that the 3 relations show exactly what is the correspondence of their sides and this gives you also a correspondence of their angles.
A: In calculating ratios of this kind, it would be easier if you divide the largest side of the larger triangle by the largest side of the smaller triangle (medium side by the medium corresponding side, etc.) such that the chance of getting round-off errors can be reduced.
Thus, $\frac{CD}{AB} = \frac {2.6}{2} = 1.3$. Similarly, you will get 1.3 for the other two sets.
After the above has been found, by the theorem “3 sides proportional”, you can claim that ⊿ABC is similar to ⊿DCA.
By “corresponding measurements”, you can further claim that $\angle BCA = \angle CAD$.
Since these two angles are in the alternate angle positions, by the theorem “alternate angles equal”, you can conclude that BC is in fact parallel to AD.
A: Continuing,
since the triangles are similar,
$x = 109$
and
$y 
= CAD
=180-(109+47)
=180-156
=24
$.
