In the diagram below DE//BC. DE = 2cm, BC = 3cm and AC = 2cm. Determine the value of x.

For the first triangle (ABC):

The angles are $57^\circ, 55^\circ, 68^\circ$.
For the second triangle (PQR):

The angles are $55^\circ, 68^\circ, 57^\circ$.

Both triangles have angles $55^\circ, 68^\circ, 57^\circ$. Since all corresponding angles are equal, the triangles are similar by the Angle-Angle-Angle (AAA) similarity criterion.

Matching angles: $\angle A = \angle R = 57^\circ$, $\angle B = \angle P = 55^\circ$, $\angle C = \angle Q = 68^\circ$.

1b)

The first triangle (ABC) has angles marked with one arc ($\angle A$) and two arcs ($\angle B$).
The second triangle (PQR) has angles marked with one arc ($\angle P$) and two arcs ($\angle Q$).
This indicates that $\angle A = \angle P$ and $\angle B = \angle Q$.

Since two angles of one triangle are equal to two angles of another triangle, their third angles must also be equal ($\angle C = \angle R$). Therefore, the triangles are similar by the Angle-Angle (AA) similarity criterion.

Matching angles: $\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$.

1c)

For the first triangle (ABC):

The angles are $40^\circ, 70^\circ, 70^\circ$.
For the second triangle (PQR):

The angles are $70^\circ, 40^\circ, 70^\circ$.

Both triangles have angles $40^\circ, 70^\circ, 70^\circ$. Since all corresponding angles are equal, the triangles are similar by the Angle-Angle-Angle (AAA) similarity criterion.

Matching angles: $\angle A = \angle Q = 40^\circ$, $\angle B = \angle P = 70^\circ$, $\angle C = \angle R = 70^\circ$.

1d)

For the first triangle (ABC):

The angles are $69^\circ, 37^\circ, 74^\circ$.
For the second triangle (PQR):

The angles are $54^\circ, 57^\circ, 69^\circ$.

The angles of $\triangle ABC$ are $69^\circ, 37^\circ, 74^\circ$.
The angles of $\triangle PQR$ are $54^\circ, 57^\circ, 69^\circ$.
The sets of angles are not identical. Only one angle ($69^\circ$) is common, but not all three.

Since the corresponding angles are not all equal, the triangles are not similar.