Mathematics • Year 7 • Unit 3 • Lesson 6
Triangle Problem Solving — Real World
Use angle sum, exterior angle, isosceles and equilateral rules to find unknown angles in bridge trusses, roof gables, sails, ramps and surveying triangles. Show every reason in brackets — markers always look for it.
1. Word problems
For each problem, identify which triangle the unknown sits in, pick the right rule, and state the reason in brackets. A bare answer with no working only earns half marks.
1.1 — Roof gable. The triangular front of a roof gable is an isosceles triangle. The apex angle (at the top of the roof) measures 80°.
(a) Find each of the two base angles of the gable.
(b) State the reason in brackets for your line of working. 2 marks
1.2 — Bridge truss. A bridge truss has triangle PQR with ∠P = 50° and ∠Q = 60°. The exterior angle at vertex R extends along the next truss panel.
(a) Use the angle sum to find the interior ∠R.
(b) Use the exterior angle theorem to find the exterior angle at R as the sum of two remote interior angles.
(c) Check that interior + exterior at R = 180°. 3 marks
1.3 — Sail design. A triangular yacht sail has three corner angles in the ratio 2 : 3 : 5. Let the three angles be 2x°, 3x° and 5x°.
(a) Write the equation and the reason.
(b) Solve for x.
(c) State the three angles of the sail. 3 marks
1.4 — Ramp at the skate park. The side view of a skate ramp is a right-angled triangle. The angle the ramp makes with the ground is 25°.
(a) Identify the third angle (between the back support and the ramp surface) using the angle sum.
(b) State the reason. 2 marks
1.5 — Surveying two stations. A surveyor sights three stations A, B and C. Triangle ABC has ∠A = 35° and ∠C = 35°.
(a) Explain why triangle ABC is isosceles.
(b) Find ∠B with reason. 2 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences and name the rule(s) you used. 4 marks
2.1 A classmate is shown a triangle with angles 40°, 70° and an unknown angle x°, and writes simply "x = 70". Explain (i) what rule they used (probably without naming it), (ii) what the correct working with a reason in brackets should look like, and (iii) why "state the reason" is worth marks even when the number is right.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Roof gable, apex 80°
(a) Each base angle = (180 − 80) ÷ 2 = 100 ÷ 2 = 50°.
(b) Reason: (base ∠s isos.) for the two equal base angles, and (∠ sum of △) for the 180° equation.
Check: 80 + 50 + 50 = 180° ✓
1.2 — Bridge truss PQR
(a) ∠R = 180 − 50 − 60 = 70° (∠ sum of △).
(b) Exterior at R = ∠P + ∠Q = 50 + 60 = 110° (ext. ∠ of △).
(c) Interior + exterior at R = 70 + 110 = 180° ✓ (∠s on str. line).
1.3 — Yacht sail (2 : 3 : 5)
(a) 2x + 3x + 5x = 180 (∠ sum of △).
(b) 10x = 180, so x = 18.
(c) Angles = 2(18) = 36°, 3(18) = 54°, 5(18) = 90°.
Check: 36 + 54 + 90 = 180° ✓
1.4 — Skate ramp
(a) 90 + 25 + ? = 180. Third angle = 180 − 90 − 25 = 65°.
(b) Reason: (∠ sum of △).
1.5 — Surveying triangle ABC
(a) ∠A = ∠C = 35°. Equal angles in a triangle force equal opposite sides, so triangle ABC has AB = BC and is therefore isosceles with B as the apex.
(b) ∠B = 180 − 35 − 35 = 110° (∠ sum of △). Check: 35 + 35 + 110 = 180° ✓
2.1 — Explain your thinking (sample response)
(i) My classmate used the angle sum of a triangle: 180 − 40 − 70 = 70°. They just didn't write the equation or the reason.
(ii) Full working should be: x = 180 − 40 − 70 (∠ sum of △), so x = 70°. Then check: 40 + 70 + 70 = 180° ✓.
(iii) "State the reason" is worth marks because in geometry the answer alone isn't a proof — anyone can guess a number. The reason in brackets shows the marker that you chose the rule deliberately and could use it again on a new diagram. Skipping the reason loses one of the available marks even when the number is right.
Marking: 1 for naming "angle sum of triangle"; 1 for correct full working; 1 for the check; 1 for the explanation of why reasons matter.