Mathematics • Year 8 • Unit 3 • Lesson 16
Quadrilateral Angles in the Real World
Use the 360° rule where it actually shows up: roof trusses, kite-flying, picture frames, sports fields and tiling patterns. Then explain your thinking in your own words.
1. Word problems
Each problem hides a quadrilateral. Identify the shape, set up the 360° equation, then solve. Show working — a single final number with no working earns only half marks.
1.1 — The picture frame. A wooden picture frame is shaped like a rectangle. The carpenter has measured three corners as 89°, 91° and 90°. The fourth corner is unmeasured but the carpenter claims the frame is a "true rectangle".
(a) Find the fourth angle.
(b) Is the frame a true rectangle (all angles exactly 90°)? Justify. 3 marks
1.2 — Kite design. A child draws a kite shape. The top angle (where the two equal short sides meet) is 100°, the bottom angle (where the two equal long sides meet) is 40°. The two side angles are equal to each other.
(a) Find the size of each side angle.
(b) Which lesson rule guarantees the two side angles must be equal? 3 marks
1.3 — Roof truss. A trapezoidal roof truss has its two base angles equal (both touch the horizontal beam). The two top angles are 120° each.
(a) Find the size of each base angle.
(b) Confirm the top and base angles on the same side add to 180° (co-interior angles between the parallel sides). 3 marks
1.4 — Soccer field corner. A groundskeeper marks out a parallelogram-shaped training zone. One angle measures 75°.
(a) State the sizes of the other three angles.
(b) What is the sum of all four angles? 3 marks
1.5 — Tiling the bathroom. A tiler is laying an irregular quadrilateral tile. Three angles are 95°, 105° and 80°. The tile is supposed to fit perfectly into a corner where three other identical tiles meet at a vertex (no gap, no overlap).
(a) Find the fourth angle of the tile.
(b) Will four identical tiles meet perfectly at a vertex around a point (sum 360°)? Use your answer to (a) to decide. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate has been asked to find the missing angle in a quadrilateral whose three known angles are 120°, 100° and 70°. They write "missing angle = 180° − (120 + 100 + 70) = 180° − 290° = −110°" and conclude "that's impossible". 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" they could do to catch the mistake. Use the phrase "splits into two triangles" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Picture frame
(a) Fourth angle = 360 − (89 + 91 + 90) = 360 − 270 = 90°.
(b) No, not a true rectangle. A true rectangle needs all four angles to be exactly 90°. Two of the measured angles are 89° and 91°, not 90°, so the frame is only an approximate rectangle.
1.2 — Kite design
(a) 100 + 40 + 2x = 360, so 2x = 220, each side angle = 110°.
(b) In a kite, the pair of opposite angles between the unequal sides are always equal — these are the two "side" angles in this question.
1.3 — Roof truss (trapezium)
(a) 120 + 120 + 2x = 360, so 2x = 120, each base angle = 60°.
(b) Check: 120 + 60 = 180° ✓. The top and bottom angles on the same side are co-interior between the two parallel sides, so they must add to 180°.
1.4 — Parallelogram training zone
(a) Opposite angle = 75°. Adjacent angles: 180 − 75 = 105° (and its opposite = 105°).
(b) Sum = 75 + 105 + 75 + 105 = 360° ✓
1.5 — Bathroom tile
(a) Fourth angle = 360 − (95 + 105 + 80) = 360 − 280 = 80°.
(b) For four identical tiles to fit around a vertex, one of the tile's angles, used four times, must total 360°. 4 × 80 = 320, 4 × 95 = 380, 4 × 105 = 420 — none equal 360°. So no, four identical tiles will NOT meet perfectly at a vertex using any single angle of this tile.
2.1 — Explain your thinking (sample response)
The classmate has used the wrong angle-sum rule. They used 180° (which is the sum for a triangle), but a quadrilateral has four angles, not three. Every quadrilateral splits into two triangles when you draw a diagonal, so the angle sum is 2 × 180° = 360°. The correct working is: missing angle = 360 − (120 + 100 + 70) = 360 − 290 = 70°. A quick sanity check would be to remember that the four angles must add to a positive number — a negative answer like −110° is a strong signal something is wrong. Another check: the three known angles already total 290°, which is bigger than 180° but smaller than 360°, so there must be room for a positive fourth angle.
Marking: 1 mark for spotting that they used 180° instead of 360°; 1 mark for showing 360 − 290 = 70°; 1 mark for the sanity check (negative answer = warning sign, or any valid check); 1 mark for clear full-sentence explanation using the phrase "splits into two triangles".