lesson
Which facts fix the triangle?
1 · Learn
Congruent triangles have equal corresponding sides and angles, even if one is turned over or rotated. Name the correspondence: if A matches D, B matches E and C matches F, then AB matches DE and angle B matches angle E. Do not compare unrelated sides just because they look horizontal.
SSS fixes all three sides. SAS fixes two sides AND the included angle between them. ASA fixes two angles and their included side; if a different corresponding side is known, find the third angle first to use the same idea. The angle sum of a triangle is 180°. These conditions fix size as well as shape.
For right triangles, RHS means a Right angle, the Hypotenuse and one corresponding Side other than the hypotenuse. A right triangle with hypotenuse 5 cm and one leg 3 cm has its other leg fixed at 4 cm. The reflected pair in the diagram are congruent by RHS. You must know the right angle and which side is the hypotenuse.
Three equal angles alone give similarity, not necessarily congruence: a 3-4-5 right triangle and a 6-8-10 right triangle have the same angles but different sizes. Two sides and a non-included angle are not a general congruence test either; depending on the data, different triangles can fit. Check a valid criterion instead of assuming any three facts will do.
2 · Worked example
Triangles ABC and DEF have angle A=angle D=40°, angle B=angle E=60°, and AB=DE=5 cm. Their third angles are both 80°, and the corresponding included sides agree, so ASA establishes congruence. The 5 cm side is important: angles alone would allow an enlarged triangle.
3 · Your turn
Two right triangles each have hypotenuse 13 cm and a corresponding leg 5 cm. Are they congruent, and what is the other leg? If two triangles instead share only angles 40°,60°,80°, is congruence guaranteed?
Check your answer
Yes, by RHS; the other leg is sqrt(13²-5²)=12 cm. Equal angles alone guarantee the same shape, not the same size, so they do not guarantee congruence.
4 · Apply your learning
Use a ruler and protractor to draw a triangle with an included 5 cm side and endpoint angles 40° and 60°. Compare with a reflected drawing. Then draw a twice-as-large version and explain which congruence fact it breaks.
Adult guidance and safety
Use labelled diagrams and equal-scale coordinate grids. Check the stated properties rather than judging a sketch by appearance. Model transformations with tracing paper; use safe classroom solids. An adult should supervise any cutting or compass work.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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