Angle facts and polygon reasoning · Higher
Введено| English | Значение |
|---|---|
| corresponding angles/ˌkɒrɪˈspɒndɪŋ ˈæŋɡlz/ | соответственные углы |
A surveyor measures one angle where a road crosses two parallel boundaries. Parallelism allows another angle to be deduced without measuring it.
- A surveyor measures one angle where a road crosses two parallel boundaries. Parallelism allows another angle to be deduced without measuring it.
- This lesson studies corresponding angles 同位角: Angles in matching positions at a transversal crossing two lines.
Choose the mathematical structure
- Angles around one point sum to 360°; angles on a straight line sum to 180°; vertically opposite angles are equal. When the crossed lines are parallel, corresponding and alternate angles are equal and co-interior angles sum to 180°. A triangle has angle sum 180°. Split an n-sided polygon into n-2 triangles to derive its interior sum (n-2)×180°.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
When are corresponding angles equal?
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Corresponding, alternate and co-interior facts are properties OF parallel lines. Without parallelism the equalities fail, which is why the reason must be named in a proof.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
If a straight-line angle is 68°, its neighbour is 112°; its vertically opposite angle is 68°. A parallel-line corresponding angle is also 68°, with the reason named. Drawing a line through a triangle vertex parallel to the opposite side transfers its two base angles by alternate-angle equality; the three angles then form a straight line and sum to 180°. A pentagon splits into three triangles, giving 540°. A regular hexagon has total 720°, each interior angle 120° and each exterior turn 60°. Exterior turns of a convex polygon sum to one full turn, 360°; a regular polygon with turn 45° has eight sides.
Angle facts and polygon reasoning
Angles around one point sum to 360°; angles on a straight line sum to 180°; vertically opposite angles are equal
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the straight-line neighbour of 68°.
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180-68=112°.
Test a tempting shortcut
- Corresponding/alternate equalities require parallel lines. Name the theorem rather than using informal letter-shape labels. An interior angle is not the same as an exterior turning angle.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Corresponding angles are equal even when the crossed lines are not parallel. This claim is false. Explain which definition or assumption it violates.
Find the interior-angle sum of a pentagon.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
(5-2)×180=540°.
Corresponding angles are equal even when the crossed lines are not parallel.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Corresponding/alternate equalities require parallel lines. Name the theorem rather than using informal letter-shape labels. An interior angle is not the same as an exterior turning angle.
Interpret a new situation
- AQA G3/G6 expects reasons and derivations. Mark the given parallelism, identify the angle positions and build a chain of justified equalities rather than reading angles from a sketch.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find one exterior angle of a regular hexagon.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
360/6=60°.
Match each angle fact to its value or rule.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Splitting an n-sided polygon into n-2 triangles derives the interior sum from the 180° triangle rule.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Angles in matching positions at a transversal crossing two lines. Choose the relationship, show the method, check its assumptions and interpret the result.