Goal and prerequisites
Recognize which circle theorem applies and combine it with triangle or right-angle facts. Review angles similarity before starting.
Method and its conditions
Mark the common chord or arc before comparing angles. The centre angle is twice the circumference angle on the same arc. Opposite cyclic-quadrilateral angles sum to 180°. Tangent-chord angles match the corresponding alternate-segment angles.
Worked example: show the decisions
If a centre angle on a minor arc is 100°, the circumference angle standing on that arc is 50°. If a cyclic quadrilateral has opposite angle 112°, the other is 68°. These statements require the specified circle configuration; a plausible sketch alone is not sufficient evidence.
Contrast: diagnose an incorrect approach
A tangent and radius meet at 90°, but a tangent and arbitrary chord generally do not. For a chord in a radius 13 circle, whose perpendicular distance from the centre is 5, half the chord is√(169−25)=12 and the full chord is 24. The perpendicular from the centre bisects the chord.
Transfer: try a changed situation
A tangent-chord angle is 42°. The alternate-segment triangle has another angle 65°. Find its remaining angle and name the two reasons.
Transfer solution and interpretation
The corresponding angle in the alternate segment is 42°. Triangle angles sum to 180°, so the remaining angle is 73°. State both the alternate-segment theorem and the triangle-angle sum.
A chord subtends an angle of 60° at the centre of a circle. An angle at the circumference stands on that same minor arc, with its vertex on the opposite major arc. Find that circumference angle in degrees.
- The centre angle is twice the circumference angle standing on the same arc.
- Using the stated quantities: 60/2 = 30.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Compare the other answer choices
Use equal angles instead of the centre-to-circumference factor 2.
Double the centre angle instead of halving it.
Select the supplementary circumference angle from the other segment.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027