Circle theorems and reasoned proofs · Higher
| English | 中文 | Pinyin |
|---|---|---|
| cyclic quadrilateral/ˈsaɪklɪk ˌkwɒdrɪˈlætərəl/ | 圆内接四边形 | yuán nèi jiē sì biān xíng |
Which angles see the same chord?
- Two observers see the same chord from the circle. Their angles are linked by a theorem rather than by the apparent size of the drawing.
- This lesson studies cyclic quadrilateral 圆内接四边形: A quadrilateral whose four vertices lie on one circle.
Choose the mathematical structure
- The angle at the centre is twice the angle at the circumference on the same arc. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to 180°. A radius is perpendicular to a tangent at contact.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines cyclic quadrilateral?
A quadrilateral whose four vertices lie on one circle.
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 central angle is 100°, the corresponding angle at the circumference is 50°. In a cyclic quadrilateral with one angle 112°, its opposite angle is 180-112=68°. A radius meeting a tangent gives 90°, even if the drawing looks oblique.
Circle theorems and reasoned proofs
The angle at the centre is twice the angle at the circumference on the same arc
Choose the theorem from the geometric configuration before calculating the angle.
Find the circumference angle corresponding to a 100° central angle.
For the same chord and corresponding arc, centre angle=twice circumference angle: 100/2=50°.
Test a tempting shortcut
- Identify the same chord and the correct arc before using a theorem. Two visible right angles do not prove a quadrilateral cyclic without a valid converse argument. A diagram need not be to scale.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every quadrilateral has opposite angles summing to 180°. This claim is false. Explain which definition or assumption it violates.
Find the angle opposite 112° in a cyclic quadrilateral.
Opposite cyclic angles sum to 180°, so 180-112=68°.
Every quadrilateral has opposite angles summing to 180°.
Identify the same chord and the correct arc before using a theorem. Two visible right angles do not prove a quadrilateral cyclic without a valid converse argument. A diagram need not be to scale.
Interpret a new situation
- Write one reason alongside each angle calculation. For the alternate-segment theorem, name the tangent and chord, then identify the angle in the opposite segment. Use auxiliary radii only when they help the proof.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the angle between a radius and its tangent in degrees.
A radius and its tangent are perpendicular at contact: 90°.
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.
A quadrilateral whose four vertices lie on one circle. Choose the relationship, show the method, check its assumptions and interpret the result.