Angles in parallel lines

When a line (transversal) crosses two parallel lines, it creates special pairs of angles. These angle relationships — alternate, corresponding and co-interior — are used to find missing angles.

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Key facts to remember

1Alternate angles are equal (Z-angles): they appear on opposite sides of the transversal.
2Corresponding angles are equal (F-angles): they appear in matching positions at each intersection.
3Co-interior angles (same-side interior or C-angles) sum to 180°.
4Vertically opposite angles are equal wherever two lines cross.
5Angles on a straight line sum to 180°; angles around a point sum to 360°.

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Worked examples

Example 1

Two parallel lines are cut by a transversal. One angle is 65°. Find the alternate angle and the co-interior angle.

Working

Alternate angle = 65° (equal, Z-angle)
Co-interior angle = 180° − 65° = 115°

AnswerAlternate angle = 65°; co-interior angle = 115°

Example 2

Find angle x: a transversal crosses two parallel lines. The corresponding angle to x is (3x − 20)°.

Working

Corresponding angles are equal: x = 3x − 20
20 = 2x
x = 10°

Answerx = 10°

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Common mistakes

✗Confusing alternate (Z) and co-interior (C) angles — alternate are equal, co-interior sum to 180°.

✗Assuming angles are alternate or corresponding without confirming the lines are parallel.

✗Forgetting to state the angle fact used — examiners require reasoning.

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Exam tips

✓Always state the reason: "alternate angles are equal" or "co-interior angles sum to 180°".

✓Mark parallel lines with arrows and label all known angles before working through the problem.

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