Name what each pair is.

a and b: alternate angles (a Z shape).

a and c: adjacent angles on a straight line.

a and d: co-interior angles (a C shape).

c and d: corresponding angles (an F shape).

Co-interior angles lie between the parallel lines, on the same side of the crossing line.

Only a and d are co-interior angles, so circle a and d.

The two marked angles sit in the same position at their own crossing: both above the parallel line and both on the right of the slanting line.

Angles in matching positions like that are corresponding angles, and corresponding angles are equal.

\(x = 41^\circ\)

The mark is for naming the rule, not for describing the picture. "Corresponding angles are equal" is enough; "F angles" on its own is not, because it names the shape rather than the rule.

Both marked angles lie between the parallel lines, but on opposite sides of the line \(PQ\). That is the Z shape, and those are alternate angles.

Alternate angles are equal, so

\(x = 67^\circ\)

"Alternate angles are equal" is the wording that earns the mark. Alternate angles are the pair that lie between the parallel lines and on opposite sides of the crossing line — check both halves of that before choosing this reason, because the pair between the lines on the same side add to \(180^\circ\) instead.

The two marked angles are at the same crossing, opposite each other through the point where the lines meet.

Vertically opposite angles are equal, so

\(x = 50^\circ\)

Worth noticing: the parallel lines play no part in this one. Two straight lines crossing anywhere make vertically opposite angles equal, whether or not anything on the diagram is parallel. The arrows are there because the rest of the diagram uses them, not because this angle needs them.

Both marked angles are between the parallel lines and both are on the same side of the slanting line. Angles in that position are co-interior angles — some books call them allied angles, and the shape is a C.

Co-interior angles add up to \(180^\circ\), so

\(x = 180 - 57 = 123\)

\(x = 123^\circ\)

This is the one reason of the four that is not "… are equal", and writing "co-interior angles are equal" loses the mark even when the arithmetic is right. Co-interior angles are between the parallel lines on the same side of the crossing line, and they add up to \(180^\circ\).

\(p\) and the angle of \(72^\circ\) are at the same crossing, opposite each other through the point.

Vertically opposite angles are equal, so \(p = 72^\circ\).

\(x\) and the angle of \(72^\circ\) are both between the parallel lines, and both on the same side of the slanting line. They are co-interior angles.

Co-interior angles add up to \(180^\circ\), so

\(x = 180 - 72 = 108\)

\(x = 108^\circ\)

The two parts have different answers on purpose: \(p\) is equal to the given angle and \(x\) is its supplement, so writing \(72\) twice cannot be right.

The two angles are co-interior, so they add up to \(180^\circ\).

\(x = 180 - 132 = 48\)

\(x = 48^\circ\)

Alternate angles and co-interior angles both lie between the parallel lines, which is why they are so easy to mix up. What separates them is the slanting line: alternate angles are on opposite sides of it, co-interior angles are on the same side.

Here both marked angles are on the same side of the slanting line, so they are co-interior, not alternate — and co-interior angles add up to \(180^\circ\) rather than being equal.

The diagram is drawn accurately, so this is also visible: \(132^\circ\) and \(48^\circ\) are plainly not the same size.

The diagram gives the angle at A and the angle at F. The angle at F is corresponding to one angle of triangle ACD, because CD is parallel to FG.

Stage 1. Angle ACD = angle F = 64°, because corresponding angles are equal.

Stage 2. The angles in triangle ACD add up to 180°, so

angle ADC = 180 - 58 - 64 = 58°

because angles in a triangle add up to 180°.

Stage 3. Angle CAD = 58° and angle ADC = 58°. Two angles are equal, so the sides opposite them are equal: CA = CD.

Triangle ACD is isosceles.

\(AD = DC\), so triangle \(ACD\) is isosceles and angle \(DAC\) = angle \(ACD\).

Angles in a triangle add up to \(180^\circ\), so the two equal angles share

\(180 - 56 = 124\)

Each one is half of that, so angle \(ACD = 124 \div 2 = 62^\circ\).

\(AB\) is parallel to \(DC\), and \(AC\) crosses them, so angle \(BAC\) and angle \(ACD\) are alternate angles. Alternate angles are equal, so angle \(BAC = 62^\circ\).

Angles in triangle \(ABC\) add up to \(180^\circ\):

\(180 - 84 - 62 = 34\)

Angle \(ACB = 34^\circ\)

The two marked angles are not a named pair, so this takes two steps. Here is one route.

Step 1. Call the angle above the lower line and to the right of the slanting line \(y\). It is vertically opposite the \(111^\circ\) angle, and vertically opposite angles are equal, so \(y = 111^\circ\).

Step 2. \(y\) and \(x\) are both between the parallel lines and both on the right of the slanting line, so they are co-interior angles. Co-interior angles add up to \(180^\circ\).

\(x = 180 - 111 = 69\)

\(x = 69^\circ\)

A different route earns the same marks: the \(111^\circ\) angle and the angle below the upper line on the same side are corresponding, so that one is \(111^\circ\) too, and it lies on a straight line with \(x\), which again gives \(x = 180 - 111 = 69\). What the marks are for is naming a rule at each step, not for the particular chain.