Mathematics • Year 8 • Unit 3 • Lesson 19
Congruence in the Real World
Use the four congruence tests (SSS, SAS, AAS, RHS) where they actually appear: roof trusses, identical tiles, factory parts, gates and ramps. Then explain your reasoning in full sentences.
1. Word problems
Each problem hides a pair of triangles. Decide which test (SSS, SAS, AAS, RHS) applies, write the congruence statement with vertices in matching order, then state any unknown the question asks for. Show your reasoning — a single "yes/no" earns only half marks.
1.1 — Identical roof trusses. A builder cuts two timber roof trusses to be identical. Each is a triangle with sides 2.4 m, 3.2 m, and 4.0 m. The builder needs to know they really are interchangeable.
(a) Which congruence test guarantees the trusses are identical?
(b) Write the congruence statement (label the first truss ABC and the second DEF).
(c) Explain in one sentence why the builder does not need to measure any angles. 3 marks
1.2 — Factory metal brackets. A factory makes L-shaped metal brackets. Each bracket is stamped so that two perpendicular arms are 6 cm and 8 cm long, meeting at a right angle. Two brackets are pulled at random off the line.
(a) Which test proves the two brackets are congruent (treat each bracket as a right-angled triangle with the third side being the diagonal from arm-tip to arm-tip)?
(b) Calculate the diagonal of each bracket using Pythagoras.
(c) Could a quality-control inspector use RHS instead? Why or why not? 3 marks
1.3 — Surveyor's diagonals. A surveyor marks out a rectangular block ABCD with diagonals AC and BD crossing at the centre point M. By a property of rectangles, AM = MC and BM = MD (diagonals bisect each other).
(a) Prove △ABM ≅ △CDM. State which test you used and list the three equal pairs.
(b) What does this tell us about AB and CD?
(c) Why would this property be useful when checking a rectangular sports field is the correct shape? 3 marks
1.4 — Wheelchair ramp design. A ramp manufacturer makes two right-angled wheelchair ramps. Each has a hypotenuse (the sloped surface) of 2.6 m and a horizontal base of 2.4 m.
(a) Which test proves the two ramps are congruent?
(b) Use Pythagoras to find the vertical rise of each ramp.
(c) The third side (the vertical rise) was not directly measured — explain why we still know the ramps are identical. 3 marks
1.5 — Mass-produced floor tiles. A tile maker prints triangular floor tiles with ∠1 = 60°, ∠2 = 70°, and the side BETWEEN those two angles equal to 15 cm. Two tiles are pulled from the box.
(a) Which test proves the tiles are congruent?
(b) What is the third angle of each tile, and why?
(c) Why is AAS enough to identify mass-produced parts even without measuring all three sides? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says: "If two triangles have two sides equal and one angle equal, that is always enough to prove them congruent — so SSA must be a real test." In your own words, explain (i) the difference between an included angle (SAS) and a non-included angle (SSA), (ii) why SSA is NOT a valid congruence test, and (iii) how you could quickly show your classmate two different triangles that share the same SSA data. Use the phrase "the angle must be between the two sides" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Identical roof trusses
(a) Test: SSS (all three pairs of sides equal: 2.4, 3.2, 4.0 m).
(b) △ABC ≅ △DEF (SSS).
(c) Once all three sides match, the triangle is fully determined — there is only one triangle with those three side lengths, so all angles automatically match too. No angle measurement is needed.
1.2 — Factory metal brackets
(a) Test: SAS — two perpendicular arms (6 cm and 8 cm) with the right angle BETWEEN them (the included angle).
(b) Diagonal² = 6² + 8² = 36 + 64 = 100, so diagonal = 10 cm (6-8-10 = 3-4-5 × 2).
(c) Yes — once the diagonal is measured, the inspector could equally use RHS (right angle + hypotenuse 10 cm + one leg 6 cm or 8 cm). SAS and RHS both work here because the triangle is right-angled.
1.3 — Surveyor's diagonals
(a) In △ABM and △CDM:
• AM = CM (diagonals bisect each other, given)
• BM = DM (diagonals bisect each other, given)
• ∠AMB = ∠CMD (vertically opposite angles)
Two sides plus the included angle → △ABM ≅ △CDM (SAS).
(b) Since the triangles are congruent, corresponding sides are equal, so AB = CD.
(c) On a sports field, checking that both pairs of opposite sides are equal AND that the diagonals bisect each other is enough to guarantee a parallelogram — a quick congruence check confirms the field is the correct shape without measuring every angle.
1.4 — Wheelchair ramps
(a) Test: RHS — right angle + hypotenuse 2.6 m + one other side (base) 2.4 m.
(b) Rise² = 2.6² − 2.4² = 6.76 − 5.76 = 1.0, so rise = 1.0 m.
(c) Once a right-angled triangle's hypotenuse and one leg are fixed, the third side is forced (by Pythagoras). So measuring two sides plus the right angle is enough — the third side cannot be anything else.
1.5 — Mass-produced tiles
(a) Test: AAS — two angles (60° and 70°) plus the corresponding included side (15 cm).
(b) Third angle = 180° − 60° − 70° = 50° in each tile (angle sum of a triangle).
(c) Two angles fix the SHAPE of the triangle (because the third angle is forced by the 180° rule), and one matching side fixes the SIZE. Together that's enough to lock in every other side and angle, so AAS alone proves congruence.
2.1 — SSA explanation (sample response)
An included angle (used in SAS) sits BETWEEN the two given sides — the angle must be between the two sides for SAS to apply. A non-included angle (SSA) sits opposite one of the sides, not between them. SSA is NOT a valid congruence test because the same SSA data can produce two genuinely different triangles — this is called the "ambiguous case". To show this quickly, draw a base line, mark a 30° angle at one end, then swing a compass with radius equal to the second given side from the OTHER end — depending on the side lengths, the arc can cross the angle's ray at TWO different points, giving two different triangles with the same SSA. Because the triangle is not uniquely determined, SSA cannot guarantee congruence.
Marking: 1 mark for clearly defining included vs non-included angle; 1 mark for stating SSA is invalid; 1 mark for the "ambiguous case" / two-triangle explanation; 1 mark for clear full-sentence writing including the required phrase.