Find the size of angles w, x, y and z, giving a reason for your answer in each case.

Angle $z$ and the given $60^\circ$ angle are alternate interior angles because they are between parallel lines and on opposite sides of a transversal. Alternate interior angles are equal.

The size of angle $z$ is $\boxed{\text{60}^\circ}$ (Reason: Alternate interior angles).

The triangle is isosceles, and angle $z$ is one of its base angles. Therefore, the other base angle (at the top-left vertex of the triangle) is also $60^\circ$. The sum of angles in a triangle is $180^\circ$.

The size of angle $y$ is $\boxed{\text{60}^\circ}$ (Reason: Sum of angles in an isosceles triangle).

Angle $x$ and angle $y$ are vertically opposite angles. Vertically opposite angles are equal.

The size of angle $x$ is $\boxed{\text{60}^\circ}$ (Reason: Vertically opposite angles).

Angle $w$ and the top-left base angle of the isosceles triangle (which is $60^\circ$) are alternate interior angles because they are between parallel lines and on opposite sides of a transversal. Alternate interior angles are equal.

The size of angle $w$ is $\boxed{\text{60}^\circ}$ (Reason: Alternate interior angles).

b)
In the diagram, PQ is parallel to RS, and PY = PZ.

The angle $140^\circ$ and the interior angle $\angle PRS$ form a linear pair on a straight line. Angles on a straight line sum to $180^\circ$.

Since PQ is parallel to RS, angle $\angle YPZ$ and angle $\angle PRS$ are alternate interior angles. Alternate interior angles are equal.

Given that PY = PZ, triangle PYZ is an isosceles triangle. The base angles opposite the equal sides are equal, so $\angle PYZ = \angle PZY$. The sum of angles in a triangle is $180^\circ$.

Therefore, the size of angle $x$ is $\boxed{\text{70}^\circ}$.

Question 3

a)
The diagram shows a trapezium with parallel top and bottom sides. The angles are given in terms of $x$. For parallel lines intersected by a transversal, consecutive interior angles sum to $180^\circ$.

The angles $2x$ (top left) and $2x + 10^\circ$ (bottom left) are consecutive interior angles.

The angles $3x - 20^\circ$ (top right) and $2x$ (bottom right) are consecutive interior angles.

The problem contains an inconsistency, as the value of $x$ differs depending on which pair of consecutive interior angles is used ($42.5^\circ$ from the left side and $40^\circ$ from the right side). Therefore, a unique value for $x$ cannot be determined from the given information.

Since a unique value for $x$ cannot be determined due to the inconsistency in the problem statement, we will provide the value derived from the left side as an example.
The value of $x$ is $\boxed{\text{42.5}^\circ}$ (Note: The problem contains an inconsistency, leading to different values of $x$ depending on the transversal used).