Mathematics • Year 10 • Unit 3 • Lesson 14
Mixed Geometry — Roof Trusses, Bridges and Garden Plots
Apply the full Lesson 14 toolkit — parallel-line angles, isosceles and parallelogram properties, congruence and similarity tests — to real Australian build-and-design contexts: a roof truss for a Queenslander house, a bridge cross-bracing pattern, a garden-bed trellis, and a kite-shaped sign for a Westfield foodcourt.
1. Word problems
For each problem: sketch the figure with all known angles or lengths marked, decide which tool (parallel-line angles, isosceles property, congruence, similarity) is the fastest, then write a short statement-plus-reason solution.
1.1 — Queenslander roof truss. A symmetric A-frame roof truss has the apex at the top. The two sloping rafters are equal, so the truss forms an isosceles triangle with the horizontal ceiling joist as the base. The apex angle is 80°.
(a) Find each base angle.
(b) State the geometric property used. 2 marks
1.2 — Bridge cross-brace. Two horizontal steel beams run parallel across the deck of a small footbridge in the Hunter Valley. A diagonal brace cuts across both, making an angle of 35° with the lower beam.
(a) Find the alternate angle the brace makes with the upper beam.
(b) Find the co-interior angle on the same side as the 35°. 3 marks
1.3 — Trellis on a garden bed. A timber trellis is built as a parallelogram ABCD with AB ∥ CD and AD ∥ BC. The diagonals AC and BD cross at point M.
(a) Prove that the diagonals bisect each other (i.e. AM = MC and BM = MD), using the parallel-line properties and a congruence test.
(b) State the congruence test used. 4 marks
1.4 — Kite-shaped Westfield foodcourt sign. A kite-shaped sign has AB = AD (top two sides equal) and CB = CD (bottom two sides equal). The diagonal AC is drawn.
(a) Prove that △ABC ≡ △ADC.
(b) Hence explain in one line why diagonal AC bisects both ∠BAD and ∠BCD. 3 marks
1.5 — Survey-line at a Newcastle building site. A surveyor sets up triangle ABC with point D on AB and point E on AC such that DE ∥ BC. She measures AD = 6 m, DB = 4 m and BC = 30 m. Find DE. State which tool you used (similarity or congruence) and why. 3 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A friend says: "Congruent and similar triangles are basically the same thing." Using the language of Lesson 14 (corresponding angles, proportional sides, scale factor k, congruence tests vs similarity tests), explain in 4-6 sentences (i) the difference, (ii) when congruence is the right tool and when similarity is, and (iii) the special value of k that makes the two ideas coincide. Include one concrete example of each (a congruence case and a similarity case) from real life — a roof, a sign, a model, anything.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Roof truss apex 80°
(a) Each base angle = (180° − 80°) / 2 = 50°.
(b) Equal rafters → isosceles triangle → equal base angles; angles in a triangle sum to 180°.
1.2 — Bridge cross-brace
(a) Alternate angle = 35° (alternate angles between parallel beams are equal).
(b) Co-interior angle = 180° − 35° = 145° (co-interior angles are supplementary).
1.3 — Trellis parallelogram, diagonals bisect
Consider △ABM and △CDM:
AB = CD (opposite sides of a parallelogram are equal).
∠ABM = ∠CDM (alternate angles, AB ∥ CD with transversal BD).
∠BAM = ∠DCM (alternate angles, AB ∥ CD with transversal AC).
Therefore △ABM ≡ △CDM (ASA / AAS).
Hence AM = CM and BM = DM (corresponding sides of congruent triangles). The diagonals bisect each other.
1.4 — Kite sign
(a) AB = AD (given), CB = CD (given), AC = AC (common). Therefore △ABC ≡ △ADC (SSS).
(b) Corresponding angles of congruent triangles are equal, so ∠BAC = ∠DAC and ∠BCA = ∠DCA. So AC bisects both ∠BAD and ∠BCD.
1.5 — Survey line, DE ∥ BC
AB = AD + DB = 6 + 4 = 10 m. △ADE ~ △ABC by AAA similarity (DE ∥ BC gives equal corresponding angles; ∠A is common).
k = AD / AB = 6/10 = 3/5.
DE = BC × k = 30 × 3/5 = 18 m.
Similarity (not congruence) is the right tool because the two triangles share an angle but the sides are not equal — they are proportional.
2.1 — Explain your thinking (sample response)
Congruent and similar triangles are not the same. Congruent triangles are identical in shape and size — every pair of corresponding sides is exactly equal, and every pair of corresponding angles is equal. Similar triangles are identical in shape only — corresponding angles are equal, but corresponding sides are proportional (related by a constant scale factor k). Use congruence when you need to show two things are exactly the same (e.g. the two halves of a kite-shaped Westfield sign, where △ABC ≡ △ADC means the sign is symmetric). Use similarity when you need to show two things are scaled versions of each other (e.g. a stick and a tree's shadow, where the small and large triangle have the same angles but different sizes). The two ideas coincide exactly when k = 1 — congruent is the special case of similar where the scale factor is 1.
Marking: 1 for distinguishing equal vs proportional sides, 1 for the right "when to use" guidance, 1 for k = 1 as the link, 1 for one congruence example AND one similarity example from real life.