Mathematics 7–10 · Year 9
Which three measurements fix a triangle: congruence tests by construction
Measurement and space
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The idea
Triangles constructed from three sides, two sides and the included angle, two angles and a side, or a right angle with the hypotenuse and a side always match when overlaid, while three angles fix only the shape and two sides with a non-included angle can give two different triangles; the same tests explain why a diagonal splits a parallelogram into two congruent triangles.
What you need
- Ruler with millimetre markings, a pair of compasses, a 180 degree protractor and a sharp pencil
- A4 paper, 4 sheets per learner
- Round-ended scissors
How to do it
- Working alone, construct triangles from these data: SSS 5.0 cm, 7.0 cm and 9.0 cm; SAS 6.0 cm and 8.0 cm with a 50 degree angle between them; two angles and a side, 40 and 60 degrees at the ends of a 7.0 cm side; RHS hypotenuse 10.0 cm, one side 6.0 cm and a right angle.
- Measure the unknown sides and angles of each triangle, cut the triangles out and overlay each one with three other learners' triangles made from the same data.
- Construct a triangle with angles 50, 60 and 70 degrees, choosing any side length, and compare it with a partner's.
- Construct from two sides and a non-included angle: draw a 40 degree angle at A with AB = 8.0 cm along one arm, swing a 6.0 cm arc from B across the other arm, and mark every crossing C.
- Construct a parallelogram with sides 8.0 cm and 5.0 cm and a 60 degree angle, cut it along one diagonal and overlay the two triangles; on a second copy draw both diagonals and measure the four parts into which they cut each other.
- Write an algorithm as a flowchart of yes-or-no questions that takes three given measurements and decides whether they fix exactly one triangle, and test it on the six triangle cases above.
What you should see
The SSS triangle has angles 33.6, 50.7 and 95.7 degrees; the SAS triangle's third side is 6.19 cm; the two-angle triangle has a third angle of 80 degrees and sides of 4.57 cm and 6.16 cm; the RHS triangle's third side is 8.0 cm. Every learner's triangle for these four sets matches the others when overlaid, within about 1 mm and 1 degree. The 50, 60, 70 degree triangles have the same shape and different sizes: similar, not congruent. In the two-sides case the 6.0 cm arc crosses the arm twice, because 6.0 cm lies between 8 sin 40° = 5.14 cm and 8.0 cm, giving two triangles with AC = 9.22 cm and AC = 3.04 cm (angle C = 59.0 or 121.0 degrees). The parallelogram's two triangles overlay exactly (SSS: they share the diagonal and opposite sides are equal), and its diagonals, 7.00 cm and 11.36 cm long, cut each other into halves of 3.50 cm and 5.68 cm. The learner knows it worked when the four test triangles overlay exactly, the non-included-angle construction produces both triangles, and the diagonal halves agree within 1 mm.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Three equal angles make two triangles congruent.
- Any three measurements fix a triangle.
- Two triangles are congruent only when they face the same way.
Safety card
Hazards
- compass points
- scissors
Controls
- compasses carried point down and stored closed
- cutting on the desk
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Properties of geometrical figures B (Path); the page carries the point establishing the 4 congruence tests for triangles (SSS, SAS, AAS and RHS), on which the whole of this entry rests; ACARA v9 states congruence a year earlier, at Year 8, in AC9M8SP01 (the conditions for congruence and similarity), AC9M8SP02 (properties of quadrilaterals established with congruent triangles, which is the parallelogram diagonal in step 5) and AC9M8SP04 (an algorithm that decides congruency), so no Year 9 v9 description states this concept and the AC9M9SP03 pointer covers only the flowchart in step 6; page read 2026-09-23, v9 records re-read 2026-09-23MA5-GEO-P-01
- Australian Curriculum v9AC9M9SP03
Sources
The pages the author read to write this activity.