Mathematics • Year 8 • Unit 3 • Lesson 15
Triangles in the Real World
Use the angle sum (180°) and exterior angle theorem where they appear: roof trusses, road signs, bridges, sail design, and isosceles steel braces.
1. Word problems
Each problem hides a triangle. Sketch it, label the known angles, identify which rule applies (angle sum / exterior / isosceles / equilateral), then solve. Every angle answer needs a brief reason.
1.1 — Roof truss. A symmetrical roof truss is an isosceles triangle. The two equal base angles (at the bottom of the roof) are each 28°. The apex angle (at the top) is the third angle.
(a) Find the apex angle.
(b) Why is this triangle classified as obtuse? 3 marks
1.2 — Yield (Give Way) road sign. The Australian Give Way road sign is an equilateral triangle pointing downward.
(a) What are the three interior angles of an equilateral triangle?
(b) If the road authority extends one side of the triangle to create an exterior angle at the bottom vertex, what is the size of that exterior angle? Give the reason. 3 marks
1.3 — Sail of a small dinghy. A triangular sail has angles in the ratio 3 : 4 : 5 (smallest : middle : largest).
(a) Let the angles be 3x, 4x, 5x. Use the angle sum to find x.
(b) State the three angles.
(c) Classify the triangle (right, acute, or obtuse?). 3 marks
1.4 — Steel bridge brace. A steel bridge contains a triangular brace. Two of its angles are known: a base angle of 65° and a top angle of 50°. The third angle (the other base angle) is unknown. The engineer extends the brace's left side to check the exterior angle.
(a) Find the third (interior) angle of the triangle.
(b) Find the exterior angle at the top vertex (formed by extending one side).
(c) Verify using the exterior angle theorem (exterior = sum of two remote interior angles). 3 marks
1.5 — Skate ramp triangle. A skate ramp has a right-angled triangular cross-section. The ramp meets the ground at an angle of 22° (the gentle slope for the rider). The other two angles are at the top (where the ramp meets a vertical wall) and at the bottom corner (where the ground meets the wall).
(a) Identify the right angle (which corner?).
(b) Find the third angle of the triangle.
(c) Is this an acute, right, or obtuse triangle? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked to find the missing angle in a triangle that has angles 50°, 60°, and x. They write "x = 50 + 60 = 110°". In your own words, explain (i) what mistake they have made, (ii) what the correct answer is and how to get it, and (iii) one quick sanity check that would have warned them their answer was wrong. Use the phrase "angle sum is 180°, not the sum of the other two" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Roof truss
(a) Apex = 180° − 28° − 28° = 124°.
(b) Because the apex (124°) is greater than 90°, the triangle is obtuse.
1.2 — Give Way road sign
(a) Three interior angles = 60°, 60°, 60° (equilateral).
(b) Exterior angle = 180° − 60° = 120° (angles on a straight line). Also matches the exterior angle theorem: sum of remote interior = 60° + 60° = 120° ✓.
1.3 — Dinghy sail (ratio 3 : 4 : 5)
(a) 3x + 4x + 5x = 12x = 180° → x = 15°.
(b) Angles: 3(15) = 45°, 4(15) = 60°, 5(15) = 45°, 60°, 75°.
(c) All angles < 90° → acute triangle.
1.4 — Steel bridge brace
(a) Third angle = 180° − 65° − 50° = 65° (so it's an isosceles brace).
(b) Exterior at top = 180° − 50° = 130°.
(c) Check: 65° + 65° = 130° ✓ (exterior = sum of two remote interior angles).
1.5 — Skate ramp
(a) Right angle is at the bottom corner where the vertical wall meets the horizontal ground.
(b) Third angle = 180° − 90° − 22° = 68° (the angle at the top, where the ramp meets the wall).
(c) Right-angled triangle (one angle is 90°).
2.1 — Explain your thinking (sample response)
The classmate has added the two known angles to find the third — but the angle sum is 180°, not the sum of the other two. The correct method is to subtract from 180°: x = 180° − 50° − 60° = 70°. A quick sanity check: if the third angle were 110°, the triangle would have angles 50°, 60°, and 110°, which sum to 220° — impossible. Every triangle must sum to exactly 180°, so the answer must be smaller than 180° − 60° = 120° AND it must make the total = 180° exactly.
Marking: 1 mark for spotting "added instead of subtracted from 180"; 1 mark for correct answer of 70°; 1 mark for the sanity-check (sum to 180°); 1 mark for using the required phrase.