Recognise congruence and use corresponding sides and angles
Recognise same shape and size regardless of orientation · Use sufficient triangle conditions · Distinguish congruence from similarity or an insufficient condition
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Similar figures
- Triangle sides and angles
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
- Recognise same shape and size regardless of orientation
- Use sufficient triangle conditions
- Distinguish congruence from similarity or an insufficient condition
Same size as well as shape
Congruent figures can be matched by rigid motions such as translation, rotation or reflection. Corresponding sides and angles then match exactly; an enlargement with factor other than 1 is not congruence.
Useful sufficient triangle conditions include three corresponding equal sides (SSS), two sides with their included angle equal (SAS), two angles and a corresponding side equal (ASA/AAS), and for right triangles an equal hypotenuse plus one corresponding leg (RHS). Identify corresponding vertices before transferring information.
Check whether the condition is sufficient
Equal angles alone establish triangle shape but can leave different sizes. Two sides and a non-included angle (SSA) are not a general congruence guarantee: a circle and ray construction can admit two different third vertices. Do not claim the included angle if a diagram gives a different angle.
0444 requires recognising congruent shapes and using them to solve problems. State a reasoned matching condition; diagram orientation or a rough visual fit alone is not evidence of identical dimensions.
Sources for this lesson
Pause and explain
Do three equal corresponding angles guarantee triangle congruence?
Worked example
Two triangles each have corresponding sides 3,4,5 but one is reflected. Are they congruent?
Show the worked solution
- All three corresponding side lengths match.
- SSS establishes congruence; reflection changes orientation without changing size or shape.
Answer Yes, by SSS; the reflection does not prevent congruence.
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.
Two triangles have corresponding sides 5,6,7 and 5,6,7. State a congruence reason.
Give me a hint
Compare all three paired lengths.
Compare my reasoning
- Every corresponding side matches.
- Use SSS, regardless of their drawn orientation.
Congruent by SSS.
Look for these in your work
- Match the three sides
- Do not depend on orientation
Two right triangles have equal hypotenuse 13 and an equal corresponding leg 5. What is the other leg and are they congruent?
Give me a hint
Use the right-angle condition and hypotenuse/leg data.
Compare my reasoning
- Other leg = √(13²−5²) = 12 in both.
- RHS establishes congruence.
Other leg 12; congruent by RHS.
Look for these in your work
- Identify the hypotenuse
- Use the right-triangle condition
Two triangles have angles 40°,60°,80°, but a corresponding side is 4 in one and 8 in the other. Are they congruent or similar?
Give me a hint
Check shape and size separately.
Compare my reasoning
- Matching angles establish similarity.
- The corresponding length factor is 2, not 1, so they are not congruent.
Similar with length factor 2; not congruent.
Look for these in your work
- Separate size from shape
- Use the corresponding side evidence
Common mistakes
- Similarity alone does not establish equal size.
- The SAS angle must be included between the two sides.
What can you explain now?
Do three equal corresponding angles guarantee triangle congruence?
Compare with the explanation
No; they fix shape but not size.
A larger scaled triangle has the same angles without matching lengths.
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Make it stick
Do three equal corresponding angles guarantee triangle congruence?
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.