Mathematics • Year 7 • Unit 3 • Lesson 9
Special Quadrilaterals — Real World
Use rhombus, square, kite and trapezium properties to find angles and sides in floor tiles, picnic kites, dump-truck side panels, school flag designs and tabletop layouts. State a reason in brackets after each line of working.
1. Word problems
Decide which special quadrilateral fits, use its properties, and state the reason in brackets after each line.
1.1 — Diamond tile. A floor tile is a rhombus with all four sides 12 cm long. One of its angles is 70°.
(a) Find the angle opposite the 70°.
(b) Find the two angles adjacent to the 70°.
(c) Check the four angles sum to 360°. 3 marks
1.2 — Picnic kite. A traditional picnic kite ABCD has AB = AD and CB = CD. The angle at the top (vertex A) is 80° and the angle at the bottom (vertex C) is 40°.
(a) Which pair of opposite angles is equal in the kite? Why?
(b) Find the two equal side-angles. 2 marks
1.3 — Dump-truck side panel. The side panel of a dump truck is a trapezium ABCD with AB ∥ DC (the top is parallel to the bottom). ∠A = 110° (at the front) and ∠B = 90° (at the back).
(a) Find ∠D (front-bottom).
(b) Find ∠C (back-bottom). 2 marks
1.4 — School flag. The school flag is a square ABCD with side 60 cm. The diagonals AC and BD are sewn in as colour stripes meeting at the centre M.
(a) What angle do the two stripes make where they meet?
(b) What angle does each stripe make with the edge of the flag? 2 marks
1.5 — Slanted tabletop. A "trapezium-shaped" coffee tabletop has AB ∥ DC. The angles at A and B (top) are both 100°.
(a) Find ∠C and ∠D.
(b) Is this an "isosceles trapezium" (legs equal)? Explain using the symmetry of the angle pattern. 3 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate looks at a kite and says "all four sides of a kite are equal, just like a rhombus." Explain (i) what is actually true about a kite's sides, (ii) what is true about a rhombus's sides, and (iii) why the kite-vs-rhombus mix-up is one of the most common errors in Year 7 quadrilateral problems.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Rhombus tile, side 12 cm, one angle 70°
(a) Opposite angle = 70° (opp. angles of parallelogram; rhombus ⊂ parallelogram).
(b) Each adjacent angle = 180 − 70 = 110° (co-int. angles, opposite sides parallel).
(c) Check: 70 + 110 + 70 + 110 = 360° ✓ (∠ sum of quad).
1.2 — Picnic kite, ∠A = 80°, ∠C = 40°
(a) The pair of equal opposite angles in a kite is the pair between the two pairs of unequal sides — that's ∠B and ∠D. (Reason: kite has one axis of symmetry through AC, which means it maps ∠B onto ∠D.)
(b) 80 + ∠B + 40 + ∠B = 360 (∠ sum of quad). 2∠B = 240, so ∠B = ∠D = 120°. Check: 80 + 120 + 40 + 120 = 360° ✓.
1.3 — Dump-truck trapezium, ∠A = 110°, ∠B = 90°
(a) ∠D = 180 − 110 = 70° (co-int. angles on leg AD, AB ∥ DC).
(b) ∠C = 180 − 90 = 90° (co-int. angles on leg BC, AB ∥ DC). Check: 110 + 90 + 90 + 70 = 360° ✓.
1.4 — Square flag, side 60 cm
(a) Diagonals of a square meet at 90° (square diagonals perpendicular).
(b) Each diagonal bisects a 90° corner, so the angle between a diagonal and a side = 45° (diagonals of square bisect corners).
1.5 — Tabletop trapezium, ∠A = ∠B = 100°
(a) ∠D = 180 − 100 = 80° (co-int. on leg AD); ∠C = 180 − 100 = 80° (co-int. on leg BC). Check: 100 + 100 + 80 + 80 = 360° ✓.
(b) Yes — it's an isosceles trapezium. The angles at the top are equal AND the angles at the bottom are equal, which by symmetry forces the two legs (AD and BC) to be equal in length. (Isosceles trapezium = trapezium with equal legs and base angles that match.)
2.1 — Explain your thinking (sample response)
(i) In a kite, the sides come in two PAIRS of adjacent equal sides — but the two pairs are usually DIFFERENT lengths (two "short" sides next to each other near one vertex, two "long" sides next to each other near the opposite vertex).
(ii) In a rhombus, ALL FOUR sides are the same length — there's only one length, not two.
(iii) The mix-up happens because both shapes "look diamond-y" from a distance, and both have axes of symmetry. The fix is to count side lengths: rhombus has 1 distinct side length (used 4 times); kite has 2 distinct side lengths (each used 2 times, adjacent). If a kite happens to have all four sides equal, it becomes a rhombus — but in general kites and rhombuses are different.
Marking: 1 for kite = 2 pairs of adjacent equal sides; 1 for rhombus = 4 equal sides; 1 for noting the look-alike trap; 1 for clear full-sentence writing.