Angle facts, congruence, similarity, quadrilaterals — applied
OCR J560 routinely sets multi-step geometry problems combining several angle/property facts. The key is stating the reason for each step.
Angle facts you must know
| Fact | Reason phrasing |
|---|---|
| Angles on a straight line sum to 180° | "Angles on a straight line" |
| Angles around a point sum to 360° | "Angles around a point" |
| Vertically opposite angles are equal | "Vertically opposite angles" |
| Angles in a triangle sum to 180° | "Angle sum of triangle" |
| Exterior angle of a triangle = sum of opposite interior angles | "Exterior angle of triangle" |
| Angles in a quadrilateral sum to 360° | "Angle sum of quadrilateral" |
| Co-interior angles sum to 180° (parallel lines) | "Co-interior angles, parallel lines" |
| Alternate angles are equal (parallel lines) | "Alternate angles, parallel lines" |
| Corresponding angles are equal (parallel lines) | "Corresponding angles, parallel lines" |
| Base angles of isosceles triangle are equal | "Base angles of isosceles triangle" |
Polygon angle facts
- Interior angle sum of n-sided polygon = (n − 2) × 180°.
- Exterior angle sum = 360° (always).
- Regular polygon interior angle = (n − 2) × 180°/n.
- Regular polygon exterior angle = 360°/n.
Similar shapes
Similar shapes have the same angles but possibly different sizes. Corresponding sides are in the same ratio.
Linear scale factor k means:
- Lengths × k.
- Areas × k².
- Volumes × k³.
Multi-step problem strategy
- Identify what you're asked to find.
- List relevant facts/diagrams.
- Work step by step, stating EACH reason.
- Show every angle marked on the diagram.
✦Worked example
In the figure, AB ∥ CD, angle BAC = 70°, angle ACD = 50°. Find angle BCD.
- Angle ACD = 50° (given).
- Angle BAC = 70° (given). And angle BAC = angle ACD' where D' is below — but wait, easier: angle ACB = angle BAC = 70° (alternate angles, AB ∥ CD)? Depends on setup. The general approach: identify alternate or co-interior pairs.
OCR mark schemes are very strict about REASONS. "70°" alone is not enough — the reason "alternate angles, parallel lines" is required for the M1.
OCR mark scheme conventions
- Each angle calculation should have a REASON in words. Loss of marks for unreasoned answers.
- Use the diagram — mark angles as you find them.
- Trignometry/Pythagoras only when geometry alone insufficient (Higher).
⚠Common mistakes
- Stating the answer without the reason — loses M1.
- Confusing alternate (Z-shape, equal) with co-interior (C-shape, sum to 180°).
- Forgetting interior + exterior of polygon = 180° always.
- Calculating polygon interior angle as 180/n (which is exterior, not interior).
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves