Mathematics • Year 7 • Unit 3 • Lesson 3
Triangles by Angles — Real World
Classify real-world triangles by angle: set squares, ramps, roof trusses, kites and surveyor sightings. Identify hypotenuses on right triangles and combine angle and side classifications (right isosceles, obtuse scalene, acute equilateral).
1. Word problems
Classify by angles AND sides (where info allows), and identify the hypotenuse on right triangles. Justify with reasons.
1.1 — 45° set square. A plastic 45° set square (the kind used in drafting) has angles 45°, 45° and 90°. (a) Classify by angles. (b) Classify by sides. (c) Combine the two names into one label. 3 marks
1.2 — Skateboard ramp. Looking at a wheelchair ramp from the side, you see a triangle. The ground makes a 90° corner with the back wall. The ramp rises at 15° to the ground. (a) What is the angle the ramp makes with the back wall (between the top of the wall and the top of the ramp)? (b) Classify the triangle by its angles. (c) Which side of the triangle is the hypotenuse? 3 marks
1.3 — Roof truss. A roof truss is built from triangular panels. One panel has angles 80°, 60° and 40°. (a) Classify by angles. (b) State whether the triangle is also isosceles, equilateral or scalene, and justify using equal/unequal angles. 2 marks
1.4 — Kite design. A diamond kite is divided down the middle by its spine into two identical triangles. Each half-triangle has angles 35°, 55° and 90°. (a) Classify by angles. (b) Classify by sides (use the angle values). (c) Which side of each half-triangle is the hypotenuse (the spine, the cross-spar, or the outer edge)? 3 marks
1.5 — Surveyor's sighting. A surveyor sights three landmarks: a fence post, a tree and a hilltop. The triangle formed has interior angles 100°, 50° and 30°. (a) Classify by angles. (b) Classify by sides. (c) Which side (opposite which angle) is the longest? 3 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate says "Every right triangle is also scalene because the hypotenuse is always longer than the other two sides." Decide whether the statement is correct or not, and explain (i) why side-equality is a separate question from having a right angle, (ii) give a specific counter-example (a triangle that is right-angled but NOT scalene), and (iii) explain why the right isosceles triangle (with angles 45°, 45°, 90°) exists. Use the terms hypotenuse, isosceles and right-angled.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 45-45-90 set square
(a) One 90° angle → right-angled. (b) Two equal angles (45°, 45°) → equal opposite sides → isosceles. (c) Combined: right isosceles.
1.2 — Ramp 15°, 90°, ?
(a) 180 − 15 − 90 = 75°. (b) One angle = 90° → right-angled. (c) The hypotenuse is the surface of the ramp itself — it is opposite the 90° corner at the back wall/ground.
1.3 — Roof truss 80, 60, 40
(a) All angles < 90° → acute-angled. (b) All three angles different → no two sides equal → scalene.
1.4 — Kite half-triangle 35, 55, 90
(a) Right-angled. (b) All angles different → scalene. (c) The hypotenuse is opposite the 90°. Since the 90° sits between the spine and the cross-spar, the side opposite is the outer edge of the kite.
1.5 — Surveyor 100, 50, 30
(a) One angle (100°) > 90° → obtuse-angled. (b) All angles different → scalene. (c) The longest side sits opposite the largest angle (100°), so it is the side opposite the 100° vertex.
2.1 — Explain your thinking (sample response)
The classmate is wrong. Side-equality (scalene vs isosceles vs equilateral) is a separate question from whether the triangle has a right angle. A right-angled triangle can absolutely have two equal sides — the classic counter-example is the right isosceles triangle with angles 45°, 45°, 90° (the kind found in a plastic set square). It has two equal short sides (the legs), and the hypotenuse opposite the right angle is longer than each leg but the two legs are still equal to each other. This triangle exists because the two equal sides force two equal opposite angles, which together with the 90° must sum to 180° — leaving each of the equal angles as 90 ÷ 2 = 45°. So a triangle can be both right-angled and isosceles at the same time.
Marking: 1 for identifying the statement as wrong; 1 for the right-isosceles counter-example; 1 for explaining the side/angle independence; 1 for clear use of all three required terms in full sentences.