Angle Properties
Angle Properties
All angle facts in geometry follow from a small set of foundational rules. Being able to state the correct rule and give a reason is essential — in proofs and explanations, every angle statement must be justified by name.
Angles at a Point, on a Line, and Vertically Opposite
| Rule | Value | Reason (as written in exams) |
|---|---|---|
| Angles on a straight line | sum to 180° | "angles on a straight line" |
| Angles at a point | sum to 360° | "angles at a point" |
| Vertically opposite angles | are equal | "vertically opposite angles" |
Parallel Lines
When a transversal crosses two parallel lines, three pairs of angle relationships arise:
| Pair | Relationship | Reason |
|---|---|---|
| Alternate angles | Equal (Z-angles) | "alternate angles, \( AB \parallel CD \)" |
| Corresponding angles | Equal (F-angles) | "corresponding angles, \( AB \parallel CD \)" |
| Co-interior angles | Sum to 180° (C-angles) | "co-interior angles, \( AB \parallel CD \)" |
Triangle and Polygon Angle Sums
- Angles in a triangle: \( 180° \)
- Exterior angle of a triangle = sum of the two non-adjacent interior angles
- Interior angle sum of an \( n \)-gon: \( (n-2) \times 180° \)
- Exterior angles of any polygon: sum to \( 360° \)
Worked Examples
Lines \( PQ \parallel RS \). A transversal makes an angle of 65° with \( PQ \). Find the alternate, corresponding and co-interior angles.
Alternate angle: 65° (alternate angles, \( PQ \parallel RS \)).
Corresponding angle: 65° (corresponding angles, \( PQ \parallel RS \)).
Co-interior angle: \( 180° - 65° = 115° \) (co-interior angles, \( PQ \parallel RS \)).
In triangle \( ABC \), \( \angle A = 48° \), \( \angle B = 73° \). Find \( \angle C \) and the exterior angle at \( C \).
\[ \angle C = 180 - 48 - 73 = 59° \] \[ \text{Exterior angle at } C = 48 + 73 = 121° \]Find the interior angle of a regular nonagon (9 sides). Find the number of sides of a regular polygon with interior angle 156°.
\[ \text{Interior sum} = 7 \times 180 = 1260° \implies \text{each} = 140° \] \[ \text{Exterior} = 180 - 156 = 24° \implies n = 360 \div 24 = 15 \text{ sides} \]Key Takeaways
- Always state the angle rule as a reason: "alternate angles", "corresponding angles", "co-interior angles", "angles on a straight line", etc.
- Alternate = equal (Z). Corresponding = equal (F). Co-interior = 180° (C).
- Exterior angle of a triangle = sum of the two remote interior angles.
- Interior sum of \( n \)-gon: \( (n-2)\times 180° \). Exterior angles always sum to 360°.