Define lines, segments, angles and circles
Distinguish lines, rays and segments · Define parallelism and perpendicularity · Relate angle measure to a fraction of a circular turn
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Points and distance
- Ratios and degrees
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Distinguish lines, rays and segments
- Define parallelism and perpendicularity
- Relate angle measure to a fraction of a circular turn
Start with the primitives
Point, line and distance are basic notions used to build these definitions. A line extends indefinitely both ways; a ray starts at one endpoint and extends one way; a segment is the part of a line between two endpoints, including those endpoints.
Distinct coplanar lines are parallel when they never meet. Perpendicular lines meet at a right angle. Lines crossing in a perspective drawing are not necessarily intersecting in three-dimensional space, so keep the plane or spatial context clear.
Circle and angle
A circle is the set of points in a plane at fixed positive distance r from a center. Points strictly nearer form its interior, not its circumference. A disk includes that interior and boundary.
An angle describes a turn between rays with a common endpoint. Specify which turn if a reflex and non-reflex angle share the rays. A circular arc that is a fraction f of a whole turn corresponds to 360f degrees; a quarter-turn is 90°. A segment's distance is along a straight line; arc distance is along the circular boundary.
Sources for this lesson
Pause and explain
Is the center a point on a circle of positive radius?
Worked example
An arc occupies one sixth of a complete circular turn. What central angle does it represent? Is the center on the circle?
Show the worked solution
- 360°/6 = 60°.
- The center has distance zero from itself, less than the positive radius.
Answer 60°; the center is inside, not on the circle.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Name a straight object extending both ways, one with one endpoint, and one with two endpoints.
Give me a hint
Use the endpoint structure.
Compare my reasoning
- Both directions indefinitely: line.
- One endpoint: ray; two included endpoints: segment.
Line, ray, segment, respectively.
Look for these in your work
- Use the definition
- Do not infer from the drawn length
A circle has radius 4. Classify points at distance 2, 4 and 6 from its center.
Give me a hint
Compare each distance with the radius.
Compare my reasoning
- 2 is less than 4: inside.
- 4 is on the boundary; 6 is outside.
Inside, on the circle, outside.
Look for these in your work
- Use the distance condition
- Distinguish boundary and interior
An arc is three eighths of a full turn. Find its central angle and the remaining turn.
Give me a hint
Use fractions of 360°.
Compare my reasoning
- 3/8 × 360° = 135°.
- Remaining turn = 360°−135° = 225°.
135° for the specified arc; remaining turn 225°.
Look for these in your work
- Identify the chosen arc
- Account for the full turn
Common mistakes
- A drawn short line need not be a finite segment.
- Circle and disk describe different sets.
What can you explain now?
Is the center a point on a circle of positive radius?
Compare with the explanation
No; its distance from itself is zero.
Points on the circle have distance exactly the positive radius.
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Make it stick
Is the center a point on a circle of positive radius?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.