When two parallel lines are crossed by a third line, called the transversal, the angles formed come in related sets. Alternate and corresponding angles are equal, and co-interior angles add to 180°. This appears in straightforward “find the angle” questions and inside larger diagrams with triangles and quadrilaterals.
It is the first lesson in angles and geometric reasoning, and the other lessons keep borrowing it.
Which angle pairs do parallel lines create?
Suppose AB is parallel to CD and a transversal cuts AB at P and CD at Q. Three pairs matter:
| Pair | Shape | Rule |
|---|---|---|
| Alternate | Z | Equal |
| Corresponding | F | Equal |
| Co-interior | C or U | Add to 180° |
Corresponding angles are in the same position at each crossing, for example both “top right”. Alternate angles are on opposite sides of the transversal, inside the parallel lines. Co-interior angles are on the same side, inside the parallel lines.
Vertically opposite angles and angles on a straight line still work too. Some questions need one of those first to reach the pair you want.
How do you use them step by step?
- Mark the parallel lines with arrows if the question states they are parallel.
- Find the angle you know and the angle you want.
- Trace the shape joining them: Z, F or C.
- Write the rule (equal or add to 180°) and the reason in words.
- Calculate and read the answer back against the diagram: is it acute or obtuse, and does that fit the picture?
Worked example
AB is parallel to CD. A line crosses AB at P and CD at Q, with P above Q. B is to the right of P, and D is to the right of Q.
Angle BPQ = 65°. Find angle PQD, and give a reason.
Step 1: Angle BPQ is between PB (going right) and PQ (going down). Angle PQD is between QD (going right) and QP (going up).
Step 2: Both angles lie between the parallel lines on the same side of PQ, so they are co-interior angles.
Step 3: Co-interior angles add to 180°, so ∠PQD = 180° − 65° = 115°.
Reason: co-interior angles add to 180°, because AB is parallel to CD.
Check: with C to the left of Q, ∠PQC is alternate to ∠BPQ, so it is 65°. Then ∠PQC + ∠PQD = 65° + 115° = 180°, which fits the straight line CD.
The mistake to watch for
A common slip is to see “parallel lines” and call every angle pair equal.
Mistaken answer: ∠PQD = 65°, because alternate angles are equal.
The student noticed the parallel lines but did not trace the shape. The two angles form a C shape, not a Z.
The correction is to trace the shape every time. A Z gives equal angles, a C gives angles that add to 180°. A quick sense check also helps: ∠PQD looks obtuse in the picture, so 65° cannot be right.
Check yourself
Try these, then open each answer.
1. AB is parallel to CD. Two co-interior angles are x and 108°. Find x.
Show answer
Co-interior angles add to 180°, so x = 180° − 108° = 72°.
2. Two alternate angles between parallel lines are (3x + 10)° and (2x + 35)°. Find x and the size of each angle.
Show answer
Alternate angles are equal: 3x + 10 = 2x + 35, so x = 25. Each angle is 3(25) + 10 = 85°, and 2(25) + 35 = 85° confirms it. x = 25, angles are 85°.
3. Two co-interior angles are 4x° and (x + 30)°. Find both angles.
Show answer
4x + x + 30 = 180, so 5x = 150 and x = 30. The angles are 4(30) = 120° and 30 + 30 = 60°. Check: 120° + 60° = 180°. 120° and 60°
Where this leads next
With parallel lines secure, move on to explaining an angle using triangle properties, where these rules combine with the angle sum. The non-calculator working trainer is handy for the arithmetic, and the mixed practice set tests the whole module.
Some students can do the sums but lose marks because the reason is missing or mislabelled. That is something our teachers look for in online one-to-one Mathematics tuition.