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This page sets out the method for angles in parallel lines in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Angles in parallel lines topic page.
When two parallel lines are crossed by a straight line and one angle is given.
When angles are labelled with , decide whether the pair is equal or adds to , then form an equation.
They sit in the same position at each intersection. They are equal.
Both above the parallel lines and both to the right of the transversal, for example.
They sit on opposite sides of the transversal, between the parallel lines. They are equal.
They form a Z shape.
They sit on the same side of the transversal, between the parallel lines.
They add to 180°. They are not equal unless each is 90°.
At a single crossing, opposite angles are equal.
Use that to move from a given angle to the one the parallel-line rule needs.
Write the rule you used: corresponding, alternate, or co-interior.
“They look equal” is not a reason.
These three rules need the parallel lines.
Without parallel lines, use the angles in a triangle or on a straight line.
Start by marking the parallel lines and the transversal. Pick out the angle you know and the angle you want. If they are in the same corner of each crossing, they are corresponding, and you copy the number. If they sit on a Z, one on each parallel line and on opposite sides of the transversal, they are alternate, and you copy the number. If they sit on a C, between the parallel lines and on the same side of the transversal, they are co-interior, and you subtract from 180^°.
Do not call an equal pair “alternate” just because it is equal. Corresponding angles are equal too. The mark scheme wants the name that matches the position. A short written reason is enough: “corresponding angles are equal” or “co-interior angles add to 180^°”. Vertically opposite angles, and angles on a straight line, still work at a single crossing. Use them to move from a top-left angle to the angle next to it before you jump to the other parallel line.
Method: 1. Mark the parallel lines and the transversal, then name the pair. 2. Copy the number for an F pair or a Z pair. 3. Subtract from 180^° for a C pair.
Corresponding angles are the F. Slide the top angle down the transversal without turning it over, and it lands on the matching angle. If the top-left angle is 65^°, the corresponding top-left angle on the lower line is 65^° as well. Nothing is subtracted. The reason is the parallel lines, not a guess that the diagram looks the same.
Alternate angles are the Z. Take the angle between the parallel lines on the left of the transversal at the top crossing. The angle between the parallel lines on the right of the transversal at the bottom crossing is equal to it. An angle of 58^° has an alternate angle of 58^°. It is easy to confuse this with the corresponding angle, because both equal 58^°. They are different angles on the diagram. Trace the Z with your pencil. If the Z is not there, you have a corresponding pair, or a co-interior pair, instead.
Method: 1. Copy a corresponding angle along the F, with no subtraction. 2. A top-left angle of 65^° matches a top-left angle of 65^° on the other line. 3. Copy an alternate angle along the Z, so 58^° gives another 58^°.
Co-interior angles are the pair that catch people out, because they are not equal. They lie between the parallel lines, on the same side of the transversal, in a C shape. Together they make a half turn, so they add to 180^°. If one of them is 72^°, the other is 180^° - 72^° = 108^°. If one of them is 116^°, the other is 180^° - 116^° = 64^°.
Allied angles are the same pair under another name. Some questions say “allied” and some say “co-interior”. Use the same subtraction. After you have found that 64^°, an alternate angle to it is also 64^°, because alternate angles are equal. That is two facts in a row: first the C-shape subtraction, then the Z-shape copy. Write both reasons. A single “angles in parallel lines” is too vague to earn the method mark.
Method: 1. Find the C pair between the parallel lines and subtract from 180^°. 2. Next to 72^°, work out 180-72=108^°. 3. Next to 116^°, work out 180-116=64^°.
Parallel-line angles often feed a triangle. Find the angle the triangle needs by naming the parallel-line fact, then use the angle sum of a triangle, which is 180^°. Suppose the alternate angle inside the triangle is 58^° and another angle of the triangle is 47^°. Those two use up 58^° + 47^° = 105^°, so the third angle is 180^° - 58^° - 47^° = 75^°.
Keep the two subtractions separate in your working. 180^° - 58^° = 122^° is the co-interior angle on the parallel lines. 180^° - 58^° - 47^° = 75^° is the spare angle in the triangle. They are different questions that happen to start from the same 58^°. Do not call 122^° an angle of the triangle. 122^° - 47^° gives the same 75^° only because it is the same subtraction in a different order, and the working would name an angle that is not in the triangle.
Method: 1. Move the angle into the triangle with a parallel-line fact, then use the triangle sum. 2. Angles 58^° and 47^° use 105^°, so the third angle is 180-58-47=75^°. 3. Keep the co-interior result 180-58=122^° separate from the triangle angle.
Exam questions on parallel lines usually say "give a reason for each stage of your working". Each stage earns a mark for the number and a mark for the reason, so you need short, correct phrases. Use "alternate angles are equal" for a Z shape, "corresponding angles are equal" for an F shape, and "co-interior angles add up to 180^°" for a C or U shape. Avoid vague words like "because they are parallel".
You also need the facts that are not about parallel lines. "Angles on a straight line add up to 180^°", "angles at a point add up to 360^°", "vertically opposite angles are equal", "angles in a triangle add up to 180^°", and "base angles of an isosceles triangle are equal". A typical answer is a short chain, with one line for each angle you find, and a reason written next to it.
Method: 1. Mark the angle you are given on the diagram in pencil. 2. Find an angle you can reach in one step and write its reason next to it. 3. Continue, each time using the angle you have just found. 4. Finish by stating the angle asked for with a final reason.
Tip: Write the reason in full, for example "alternate angles are equal", not just "alternate".
Higher tier questions label angles with expressions such as (3x + 10)^° and (5x - 20)^°. First decide which fact links the two angles. If they are alternate or corresponding, they are equal, so set the expressions equal and solve. If they are co-interior, they add up to 180^°, so write (3x + 10) + (5x - 20) = 180 and solve. Choosing the wrong fact is a frequent source of lost marks.
After you find x, go back and work out the actual angle. Many questions ask for the angle and not for x. Substitute your value into both expressions as a check. If the two angles should be equal, they must give the same number. If they should total 180^°, add them. A negative or very large x often means you chose the wrong fact.
Method: 1. Name the angle relationship: alternate, corresponding or co-interior. 2. Write an equation: equal for alternate or corresponding, total 180 for co-interior. 3. Solve for x, then substitute back to find the angle. 4. Check the angles satisfy the rule you used.
Harder questions put a triangle between two parallel lines. You then need the triangle facts as well as the parallel line facts. For example, if two sides of the triangle are equal, the base angles are equal. A good habit is to find any angle on the parallel lines first, then move it into the triangle with an alternate angle or a straight line, and then use the triangle angle sum.
In a quadrilateral like a trapezium, the two parallel sides make co-interior angles with each slanting side. So in a trapezium with AB parallel DC, the angles at A and D add up to 180^°, and so do the angles at B and C. Check your work by adding all four angles: the total must be 360^°. This catches many small arithmetic slips.
Method: 1. Pick a given angle and move it with a parallel line fact. 2. Use straight lines and triangle facts to find the next angle. 3. Use the isosceles fact when two sides are marked equal. 4. Check totals: 180^° for a triangle and 360^° for a quadrilateral.
Tip: In a trapezium, the angles between the parallel sides and a slanting side always add up to 180^°.
A proof such as "prove that opposite angles of a parallelogram are equal" is a chain of parallel line facts, with a reason for each line. State what you know, apply a fact, state the result, and finish with the statement you were asked to prove. Write the full sentence for the last line, not just a number.
Bearings are also a parallel lines topic. North lines drawn at different points are parallel, so co-interior and alternate angles link a bearing at one point to the bearing back at the other. If B is on a bearing of 072^° from A, the co-interior angle at B is 180 - 72 = 108^°, so the bearing of A from B is 360 - 108 = 252^°. This is the "add or subtract 180" rule explained.
Method: 1. Draw the relevant parallel lines, or north lines for a bearing. 2. Label the angle you know and the angle you want. 3. Choose the fact that links them and write it as a sentence. 4. End with the result stated clearly.
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