Mathematics • Year 7 • Unit 3 • Lesson 14

Congruent Figures — Real World

Roof trusses, mass-produced products, sandwich halves, kite spars and tiling patterns rely on triangles being identical. Use SSS/SAS/AAS/RHS to prove pairs of real triangles are congruent.

Apply · Real-World Maths

1. Word problems

For each scenario: list the equal parts, name the test, and write the congruence statement.

1.1 — Roof truss. A roof truss has two diagonal beams forming triangles on the left and right halves. Each triangle has a vertical post 2 m tall, a horizontal beam 3 m long, and a diagonal beam connecting their ends. The two triangles share the central post, so the post (2 m) is common to both, the horizontal beams are both 3 m, and the angle at the bottom between post and horizontal is 90° for both. Are the two triangles congruent? Which test? 2 marks

Stuck? Two sides (2 m and 3 m) + the angle BETWEEN them (90°). That's SAS.

1.2 — Sandwich halves. A rectangular sandwich is cut along its diagonal into two triangular halves. In each half, the two perpendicular sides are 12 cm and 8 cm, with a 90° angle between them. (a) Are the two halves congruent? (b) Which test? 2 marks

Stuck? Same two sides + included right angle = SAS. (Or right angle + hypotenuse + side = RHS.)

1.3 — Kite spars. A diamond kite has two perpendicular spars (sticks) crossing at the centre. The vertical spar is 60 cm long, the horizontal spar is 40 cm long, and they cross such that 30 cm of vertical is above and 30 cm below; 20 cm of horizontal is left and 20 cm right. The kite skin is divided by the spars into four triangles. Compare the TOP-LEFT and TOP-RIGHT triangles. (a) List two sides and the included angle they share. (b) Are they congruent? (c) Which test? 3 marks

Stuck? Each top triangle has sides 30 cm (vertical) and 20 cm (horizontal) meeting at 90° in the centre. SAS.

1.4 — Mass-produced bracket. A factory makes triangular metal brackets. Quality control measures one bracket: sides 10 cm, 24 cm, 26 cm. A second bracket is measured: sides 10 cm, 24 cm, 26 cm. Are the two brackets congruent? Which test? 2 marks

Stuck? All three pairs of sides equal = SSS.

1.5 — Ladder against a wall. Two 5 m ladders are propped against a vertical wall on level ground. Both ladders form a 70° angle with the ground. Each ladder + wall + ground forms a triangle. Are the two triangles congruent? Which test? 2 marks

Stuck? Each ladder triangle has the right angle (90° at wall–ground), the 70° angle (ladder–ground) and the 5 m side (the ladder = hypotenuse). Two angles + the side opposite the unknown angle = AAS. Or: the third angle is also 20° in both, then AAS confirms.

2. Explain your thinking

Full sentences. 4 marks

2.1 A student claims: "If two triangles have the same area, they are congruent." Is this claim correct? In a short paragraph: (i) state whether the claim is true or false, (ii) explain WHY using the meaning of congruence from Lesson 14 (same shape AND same size), and (iii) give one specific example of two triangles with the same area that are NOT congruent.

Stuck? A 2 cm × 6 cm right triangle and a 3 cm × 4 cm right triangle both have area 6 cm² but very different shapes.

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Roof truss

Equal parts: post 2 m, angle 90° between post and horizontal, horizontal beam 3 m.
Two sides + included angle = SAS. Yes — the two triangles are congruent (SAS).

1.2 — Sandwich halves

(a) Yes. (b) Two sides (12 cm and 8 cm) with the included 90° angle → SAS. (Also valid: right angle + hypotenuse + one side → RHS, since the diagonal is the hypotenuse and the same in both halves.)

1.3 — Kite spars (top-left vs top-right)

(a) Each top triangle has: vertical side 30 cm, horizontal side 20 cm, with the included angle 90° at the centre.
(b) Yes, congruent.
(c) Two sides + included angle = SAS.

1.4 — Mass-produced brackets

All three sides match (10, 24, 26) → SSS. Yes, the brackets are congruent.

1.5 — Two ladders

Each triangle has: 90° at wall-ground (right angle), 70° at ladder-ground, 5 m ladder (the hypotenuse). Two angles + a matching side → AAS. Yes, the triangles are congruent. (Alternatively, the third angle in each = 180 − 90 − 70 = 20°, and the hypotenuse = 5 m: hypotenuse + right angle + matching angle would also justify AAS.)

2.1 — Same area ≠ congruent (sample response)

The claim is false. Congruent triangles must have the SAME shape AND the SAME size — meaning all three pairs of corresponding sides are equal AND all three pairs of corresponding angles are equal. Two triangles can have the same area without matching shapes, because area depends only on base × height ÷ 2, not on the specific side lengths or angles. For example: a right triangle with legs 2 cm and 6 cm has area (1/2)(2)(6) = 6 cm², and a right triangle with legs 3 cm and 4 cm also has area (1/2)(3)(4) = 6 cm². Both have area 6 cm² but their sides (2, 6, √40) vs (3, 4, 5) are clearly different — so they are NOT congruent. Same area does not imply congruence; you need one of the four tests (SSS, SAS, AAS, RHS) to prove congruence.

Marking: 1 for "false"; 1 for the definition of congruence; 1 for a valid counter-example; 1 for clear full-sentence explanation.