Find the value of x, y, and z.

The angles $x$, $2x$, and $3x$ lie on the straight line ABC. The sum of angles on a straight line is $180^\circ$.

Reason: Angles on a straight line.

The value of $x$ is $\boxed{\text{30}^\circ}$.

The lines PQ and RS are parallel (indicated by the arrows). The other two lines are transversals.

Angle $y$ and the $30^\circ$ angle (at P) are alternate interior angles. Since PQ is parallel to RS, alternate interior angles are equal.

Reason: Alternate interior angles (PQ $\parallel$ RS).

Angle $z$ and the $42^\circ$ angle (at Q) are alternate interior angles. Since PQ is parallel to RS, alternate interior angles are equal.

Reason: Alternate interior angles (PQ $\parallel$ RS).

Consider the triangle formed by the intersection of the two transversals and the line segment RS. The angles at R and S within this triangle are $y$ and $z$. The angle vertically opposite to $x$ is the third angle in this triangle. The sum of angles in a triangle is $180^\circ$.
Let the angle at the intersection point inside the triangle be $\theta$.

Angle $x$ is vertically opposite to $\theta$. Vertically opposite angles are equal.

Reason: Vertically opposite angles and sum of angles in a triangle.

The sizes are: $x = \boxed{\text{108}^\circ}$, $y = \boxed{\text{30}^\circ}$, $z = \boxed{\text{42}^\circ}$.

The angle $c$ and the $106^\circ$ angle are vertically opposite angles.

Vertically opposite angles are equal.

Reason: Vertically opposite angles.

The unknown angle $c$ is $\boxed{\text{106}^\circ}$.