Find the sizes of the angles marked with letters. Set out your reasons

The angle $x^\circ$ and the angle $60^\circ$ are alternate interior angles.

Reason: Alternate interior angles.
The value of $x$ is $\boxed{\text{60}}$.

The exterior angle $120^\circ$ is equal to the sum of the two opposite interior angles, $x^\circ$ and $40^\circ$.

This angle is part of the large triangle.

The angles are $y^\circ$, $(30^\circ + 40^\circ)$, and $60^\circ$.

Reason: Exterior angle of a triangle equals the sum of opposite interior angles. Angles on a straight line sum to $180^\circ$. Sum of angles in a triangle is $180^\circ$.
The values are $x = \boxed{\text{80}}$ and $y = \boxed{\text{50}}$.

The angles are $x^\circ$, $30^\circ$, and $30^\circ$.

Reason: Sum of angles in a triangle is $180^\circ$.
The value of $x$ is $\boxed{\text{120}}$.

This divides $x^\circ$ into two parts, $x_1^\circ$ (upper part) and $x_2^\circ$ (lower part).

The angle $120^\circ$ and the interior angle adjacent to it form a linear pair. This interior angle is $180^\circ - 120^\circ = 60^\circ$. This $60^\circ$ angle is alternate interior to $x_1^\circ$.

The angle $30^\circ$ is alternate interior to $x_2^\circ$.

Reason: Angles on a straight line sum to $180^\circ$. Alternate interior angles are equal when lines are parallel.
The value of $x$ is $\boxed{\text{90}}$.

The angle $40^\circ$ in the lower triangle is vertically opposite to an angle in the upper triangle. Let this angle be $V$.

The angles are $90^\circ$, $x^\circ$, and $V^\circ$.

Reason: Vertically opposite angles are equal. Sum of angles in a triangle is $180^\circ$.
The value of $x$ is $\boxed{\text{50}}$.

Let this angle be $A$.

The angle $A$ is vertically opposite to one of the angles in the right triangle. Let this angle be $B$.

The angles are $55^\circ$, $x^\circ$, and $B^\circ$.

Reason: Sum of angles in a triangle is $180^\circ$. Vertically opposite angles are equal.
The value of $x$ is $\boxed{\text{60}}$.