Circle angles and proof chains · Higher
| English | 中文 | Pinyin |
|---|---|---|
| angle at the centre/ˈæŋɡl æt ðə ˈsentə/ | 圆心角 | yuán xīn jiǎo |
A circle theorem connects angles standing on one chord. The same-looking angle on the other side of the chord can instead be supplementary.
- A circle theorem connects angles standing on one chord. The same-looking angle on the other side of the chord can instead be supplementary.
- This lesson studies angle at the centre 圆心角: An angle formed by two radii meeting at the centre.
Choose the mathematical structure
- The central angle on an arc is twice a circumference angle standing on that same arc. A diameter therefore gives a 90° circumference angle. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to 180°. State the chord, arc and segment before applying a theorem.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines angle at the centre?
An angle formed by two radii meeting at the centre.
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.
Let A,B,C lie on the circle and let O be the centre inside angle ACB. Put α=angle ACO and β=angle OCB. Equal radii make triangles AOC and BOC isosceles. Thus angle AOC=180-2α and angle COB=180-2β. Angles around O give the remaining angle AOB=360-(180-2α)-(180-2β)=2(α+β)=2 angle ACB. Other centre positions need the corresponding subtraction of isosceles angles, with the same result for the chosen arc. If AB is a diameter, angle AOB=180°, so angle ACB=90°. Two circumference angles on the same chord in the same segment each equal half the same central angle, so they agree. For opposite cyclic angles, their arcs together make 360°; half-arc angles therefore sum to 180°. A central angle 100° gives 50° at the circumference; an angle opposite 112° in a cyclic quadrilateral is 68°.
Circle angles and proof chains
The central angle on an arc is twice a circumference angle standing on that same arc
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find a circumference angle standing on the same arc as a 100° central angle.
Half the central angle is 50°.
Test a tempting shortcut
- Distinguish the reflex central angle from the smaller one and identify which arc does not contain the circumference vertex. Opposite segments can give supplementary rather than equal angles. A theorem must be tied to named points.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Any two circumference angles are equal regardless of their chords and segments. This claim is false. Explain which definition or assumption it violates.
Find the angle subtended by a diameter at the circumference.
Half of 180° is 90°.
Any two circumference angles are equal regardless of their chords and segments.
Distinguish the reflex central angle from the smaller one and identify which arc does not contain the circumference vertex. Opposite segments can give supplementary rather than equal angles. A theorem must be tied to named points.
Interpret a new situation
- AQA G10 Higher requires application and proof. Draw auxiliary radii, use isosceles base angles and point sums, then extend the proof to the intended configuration rather than inferring equality from the drawing.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the opposite cyclic angle to 112°.
Opposite angles sum to 180°: 180-112=68°.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.
An angle formed by two radii meeting at the centre. Choose the relationship, show the method, check its assumptions and interpret the result.