Circle theorems and reasoned proofs · Extension
הוצג| English | משמעות |
|---|---|
| cyclic quadrilateral/ˈsaɪklɪk ˌkwɒdrɪˈlætərəl/ | מעוגן ארבע-צלעות |
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.
The method
- 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.
A cyclic quadrilateral has one angle of 112°. Why is the opposite angle 68°?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The 180° property belongs to CYCLIC quadrilaterals - vertices on one circle - and fails for a general quadrilateral. Here 180-112=68°.
A worked case
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
For the same chord and corresponding arc, centre angle=twice circumference angle: 100/2=50°.
Where it goes wrong
- 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.
Not true: Every quadrilateral has opposite angles summing to 180°. Opposite angles sum to 180° only in a cyclic quadrilateral, one whose vertices lie on a circle.
Find the angle opposite 112° in a cyclic quadrilateral.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Opposite cyclic angles sum to 180°, so 180-112=68°.
Every quadrilateral has opposite angles summing to 180°.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Opposite angles sum to 180° only in a cyclic quadrilateral, one whose vertices lie on a circle.
Applying it
- 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.
Find the angle between a radius and its tangent in degrees.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
A radius and its tangent are perpendicular at contact: 90°.
Match each circle theorem to its statement.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Each theorem needs its correct configuration: the same arc, four concyclic vertices, or the tangent contact point.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.3.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle.
- The angle at the centre is twice the angle at the circumference on the same arc.
- Watch for: identify the same chord and the correct arc before using a theorem.
Write from a circumference angle of 70 degrees
- A central angle is 140 degrees. One angle of a cyclic quadrilateral is 125 degrees.
- Find the angle at the circumference and the opposite cyclic angle.
A central angle is 140 degrees. What is the angle at the circumference on the same arc? If a cyclic quadrilateral has an angle of 125 degrees, what is the opposite angle?
140 divided by 2 is 70, and 180 minus 125 is 55.
What is the angle at the circumference?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
A circumference angle is half the central angle on the same arc, so 140 degrees gives 70 degrees. Other options — 50 degrees (the lesson example's result for a 100 degree central angle); 140 degrees (the central angle left unchanged).