Define angles, triangles and quadrilaterals precisely
Classify angles using exact boundaries · Use side and angle properties to define plane shapes · Distinguish regularity, similarity and congruence
Section Paper 2 · Historical Extended · non-calculator
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
- Degree measure
- Equal and parallel lines
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
- Classify angles using exact boundaries
- Use side and angle properties to define plane shapes
- Distinguish regularity, similarity and congruence
Angles and polygons
An acute angle is between 0° and 90°; a right angle is 90°; an obtuse angle is between 90° and 180°; a reflex angle is between 180° and 360°. Straight and full-turn angles are exactly 180° and 360°. A pentagon, hexagon and octagon have five, six and eight sides.
An equilateral triangle has all three sides equal. Use the inclusive definition of isosceles, at least two equal sides, unless a task explicitly specifies exactly two. A regular polygon has equal sides and equal interior angles; equal sides alone do not guarantee regularity.
Read the defining conditions
A parallelogram has two pairs of parallel opposite sides; a rectangle has four right angles; a rhombus has four equal sides; a square has both properties. Thus a square is also a rectangle, rhombus and parallelogram.
For trapezoids and kites, conventions vary. These practice questions define a trapezoid as exactly one pair of parallel opposite sides, and a kite as two pairs of equal adjacent sides with unequal pair lengths. Use a question's stated convention rather than assuming that every source includes or excludes special cases in the same way. Congruent shapes have the same size and shape; similar shapes have the same shape with corresponding lengths in one common ratio.
Sources for this lesson
Pause and explain
Is every rhombus a regular quadrilateral?
Worked example
Classify 90°, 135° and 220°. Is a non-square rhombus a regular quadrilateral?
Show the worked solution
- 90° is right, 135° is obtuse and 220° is reflex.
- A non-square rhombus has equal sides but not four equal angles, so it is not regular.
Answer Right, obtuse, reflex; the non-square rhombus is not regular.
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.
Classify 45°, 180° and 300°, and name polygons with 5 and 8 sides.
Give me a hint
Use exact angle boundaries and side counts.
Compare my reasoning
- 45° is acute; 180° is straight; 300° is reflex.
- Five sides give a pentagon; eight give an octagon.
Acute, straight, reflex; pentagon and octagon.
Look for these in your work
- Treat boundaries separately
- Use exact vocabulary
A quadrilateral has four equal sides and four right angles. Give all applicable names among square, rectangle, rhombus and parallelogram.
Give me a hint
A special case can satisfy several definitions.
Compare my reasoning
- Equal sides and right angles make it a square.
- It also satisfies the definitions of rectangle, rhombus and parallelogram.
All four names apply.
Look for these in your work
- Use properties rather than appearance
- Allow overlapping families
For this question, a trapezoid has exactly one pair of parallel opposite sides. Is a rectangle a trapezoid? Would an at-least-one-pair convention change that answer?
Give me a hint
Separate the two stated conventions.
Compare my reasoning
- A rectangle has two pairs, so it fails the exactly-one condition.
- It satisfies the at-least-one condition.
No under exactly one pair; yes under at least one pair.
Look for these in your work
- State the convention
- Do not present either convention as universal
Common mistakes
- Boundary angles are not acute, obtuse or reflex.
- Read any inclusive or exclusive shape convention.
What can you explain now?
Is every rhombus a regular quadrilateral?
Compare with the explanation
No; equal sides alone do not ensure equal angles.
Only its square special case has four equal right angles.
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Make it stick
Is every rhombus a regular quadrilateral?
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.