Mathematics • Year 7 • Unit 3 • Lesson 4
Angle Sum of Triangles — Real World
Apply the 180° rule to real triangles — sails, billiard cues, roof trusses, navigation bearings and surveyor sightings. Combine angle sum with the isosceles base-angle rule, and set up simple algebraic equations.
1. Word problems
Set up the equation, solve, and always state (angle sum of △) as your reason.
1.1 — Triangular sail. A triangular sail has a bottom angle (at the boom) of 25° and a top angle (at the mast tip) of 110°. Find the third angle (at the rear bottom corner). 2 marks
1.2 — Billiards bank shot. A cue strikes a ball, which bounces off a cushion and ends at a pocket. The path forms a triangle. The angle at the ball's start point is 35° and the angle at the cushion is 90°. Find the angle at the pocket. 2 marks
1.3 — A-frame roof. The cross-section of an A-frame roof is an isosceles triangle. The apex angle (at the peak of the roof) is 110°. Find each base angle (where the roof meets the walls). 2 marks
1.4 — Garden bed corner. A triangular garden bed has angles (3x)°, (2x)° and (x + 30)°. (a) Set up an equation from the angle sum. (b) Solve for x. (c) State all three angles. 3 marks
1.5 — Surveyor's triangle. A surveyor measures two angles of a survey triangle as 42° and 67°. (a) Find the third angle. (b) Classify the triangle by angles. (c) Could the survey triangle also be isosceles? Justify your answer using the three angles. 3 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate is asked for the third angle in a triangle with two known angles of 95° and 90°. They write "180 − 95 − 90 = −5°. The answer is negative 5°." Explain (i) what mathematical mistake they have actually made — is it arithmetic, or is something else wrong? (ii) why the given angles cannot be the two angles of a valid triangle, (iii) what total the two known angles can sum to at most, and (iv) what they should write instead of "−5°". Reference the angle sum rule in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Triangular sail 25°, 110°, ?
180 − 25 − 110 = 45° (angle sum of △).
1.2 — Bank shot 35°, 90°, ?
180 − 35 − 90 = 55° (angle sum of △). (Right-angled scalene.)
1.3 — A-frame roof, apex 110°
Let each base angle = B. 110 + 2B = 180 → 2B = 70 → B = 35°. Each base angle is 35° (angle sum of △).
1.4 — Garden bed (3x)°, (2x)°, (x + 30)°
(a) 3x + 2x + (x + 30) = 180 (angle sum of △).
(b) Combine: 6x + 30 = 180 → 6x = 150 → x = 25.
(c) Angles: 3(25) = 75°, 2(25) = 50°, 25 + 30 = 55°. Triangle has angles 75°, 50°, 55°. Check: 75 + 50 + 55 = 180° ✓.
1.5 — Surveyor 42°, 67°, ?
(a) Third = 180 − 42 − 67 = 71° (angle sum of △).
(b) All angles < 90° → acute-angled.
(c) The three angles are 42°, 67°, 71°. No two are equal, so the triangle is not isosceles — it is scalene.
2.1 — Explain your thinking (sample response)
The arithmetic itself is actually correct — 180 − 95 − 90 really does equal −5. The classmate's mistake is conceptual: they failed to notice that the two given angles (95° and 90°) already sum to 185°, which is more than the total angle sum of 180° for any triangle. That means no valid triangle can exist with both of these angles, so the question cannot be answered as posed. The two known angles of a valid triangle can sum to at most just under 180° (because the third angle must be positive and greater than 0°). Instead of writing "−5°", the student should write something like: "No triangle with these two angles is possible, because 95° + 90° = 185° > 180° already, violating the angle sum rule (angle sum of △ = 180°)."
Marking: 1 for naming the conceptual (not arithmetic) error; 1 for citing 180° angle sum; 1 for "<180°" upper limit on two known angles; 1 for clear full-sentence corrected response.