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Angles on parallel lines

 

Rotate the transversal and watch which angles stay equal and which add up to 180° — one known angle is enough to work out all the others without a protractor.

Formula & notes

corresponding: ∡1 = ∡5  ·  alternate: ∡3 = ∡6

co-interior: ∡3 + ∡5 = 180°

corresponding — the same position at both intersections → equal

alternate — opposite sides of the transversal → equal

co-interior — between the parallel lines, same side → sum 180°

vertically opposite — opposite each other at one intersection → equal

linear pair — next to each other on a straight line → sum 180°

What is it?

When two parallel lines are cut by a third line — the transversal — eight angles appear. Although there are eight of them, they take only two different sizes: some angle α and its supplement 180° − α. That is why it is enough to measure or to know a single angle and all the others can be worked out logically, without a protractor.

The key is the pairs: corresponding angles lie at both intersections in the same position (for example both upper right) and they are equal. Alternate angles lie on opposite sides of the transversal and they are equal as well. Co-interior angles lie between the parallel lines on the same side of the transversal and their sum is 180°. Careful — these relationships hold only for parallel lines; if the lines are not parallel, corresponding angles will not be equal.

Calculator
°
Acute angle α
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Supplement 180° − α
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PairTypeRelation

Interactive graph

Grab the orange point on the transversal and rotate it — or type the angle into the calculator. The buttons switch which pair is highlighted.

corresponding alternate co-interior vertically opposite none
p q r
1. Corresponding angles
  • Where to look for them: at both intersections in the same position — for example both of them upper right.
  • Relation: on parallel lines they are equal: ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, ∠4 = ∠8.
  • How to remember them: they form the letter F (rotated or mirrored as well).
  • Careful: this holds only if the lines are parallel.
2. Alternate angles
  • Where to look for them: on opposite sides of the transversal.
  • Interior (between the parallel lines): ∠3 = ∠6, ∠4 = ∠5.
  • Exterior (outside the parallel lines): ∠1 = ∠8, ∠2 = ∠7.
  • Tip: alternate interior angles form the letter Z (or N).
3. Co-interior angles and linear pairs
  • Co-interior: between the parallel lines on the same side of the transversal — ∠3 + ∠5 = 180°, ∠4 + ∠6 = 180°.
  • Tip: they form the letter C (or U).
  • Linear pair: two angles lying next to each other on one straight line add up to 180°.
  • Vertically opposite: opposite each other at one intersection — they are equal.
Worked example
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Frequently asked questions
What are corresponding angles?
They lie at both intersections in the same position with respect to the transversal and to the parallel line (for example both of them upper right). On parallel lines they have the same size and they form the shape of the letter F.
What are alternate angles?
They lie on opposite sides of the transversal. The interior ones are between the parallel lines (shape of a Z), the exterior ones lie outside them. On parallel lines they are equal.
Why are co-interior angles not equal?
Because they lie on the same side of the transversal between the parallel lines — together they make a straight angle, so their sum is 180° rather than them being equal.
Why is one angle enough?
On parallel lines all eight angles take only two different sizes: α and 180° − α. So once we know one of them, the rest can be worked out without measuring with a protractor.
Practice problems

Solve it step by step — the next step unlocks only after a correct answer. You can type decimals with a dot or a comma.



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