Angle properties: at a point, on a straight line, with parallel lines
This topic underpins almost every geometry question. Learn the rules, learn the names — examiners often demand you state the rule when justifying answers.
Angles at a point
The angles around a single point sum to 360°.
Worked example: angles 145°, 60°, x and 90° meet at a point. Find x.
- 145 + 60 + 90 = 295.
- x = 360 − 295 = 65°.
Angles on a straight line
Angles on a straight line sum to 180°.
Worked example: two angles, 38° and y, lie on a straight line. y = 180 − 38 = 142°.
Vertically opposite angles
When two straight lines cross, the opposite (vertical) angles are equal.
Parallel lines and a transversal
Two parallel lines cut by a transversal make four key angle pairs:
- Corresponding angles (F-shape) — equal.
- Alternate angles (Z-shape) — equal.
- Co-interior (allied / C-shape) — sum to 180°.
- Vertically opposite — equal.
Reasons in answers
When asked to give a reason, use exactly the right phrase:
- "Angles on a straight line sum to 180°."
- "Angles around a point sum to 360°."
- "Vertically opposite angles are equal."
- "Corresponding angles are equal (parallel lines)."
- "Alternate angles are equal (parallel lines)."
- "Co-interior angles sum to 180° (parallel lines)."
✦Worked example
In a diagram, AB ∥ CD and a transversal meets them. The angle on AB is 70° (above the transversal). Find the alternate angle on CD.
- Alternate angles are equal → 70°.
⚠Common mistakes
- Using "Z, F, C" only — examiners want you to name the rule (alternate, corresponding, co-interior).
- Treating co-interior as equal — they sum to 180°, NOT equal.
- Misidentifying which angles are corresponding — they're in the same position relative to the transversal.
- Forgetting to state the parallel-line condition — co-interior, alternate, corresponding rules require parallel lines.
- Using straight-line and parallel-line rules together sloppily — show each step.
➜Try this— Quick check
In parallel lines crossed by a transversal: one angle is 110°. The co-interior angle on the other parallel line is what?
- 180 − 110 = 70°.
AI-generated · claude-opus-4-7 · v3-deep-geometry