Mathematics • Year 7 • Unit 3 • Lesson 7

Quadrilaterals in the Real World

Spot quadrilaterals in tiles, windows, road signs, kites and tabletops. Use the 360° angle sum to find a missing angle, and name each shape with the MOST specific name from the family tree.

Apply · Real-World Maths

1. Word problems

For each scenario: (i) identify which special quadrilateral fits the description, and (ii) if there's an angle to find, use the 360° rule with a reason in brackets.

1.1 — Bathroom tile. A bathroom tile has 4 equal sides AND 4 right angles.

(a) Name the tile using its most specific name.
(b) List every other quadrilateral category it ALSO belongs to. 2 marks

Stuck? Most specific = square; then trace up the family tree.

1.2 — Picture frame. A picture frame has 4 right angles, with the two short sides 20 cm long and the two long sides 30 cm long.

(a) Why is it NOT a square?
(b) What is its most specific name?
(c) List the other category it belongs to. 2 marks

Stuck? Square needs ALL four sides equal — these are 20, 30, 20, 30.

1.3 — Children's kite. A traditional kite shape has two short adjacent sides equal in length, and two long adjacent sides equal in length. No sides are parallel.

(a) Name the shape.
(b) Three of its angles are 100°, 100° and 60°. Find the fourth angle and state the reason in brackets. 2 marks

Stuck? Use 360° − sum of the three known angles.

1.4 — Road sign. A road sign has exactly one pair of parallel sides. Its other pair of sides is slanted (not parallel).

(a) Name the sign using the NSW Stage 4 definition.
(b) If three of its angles are 70°, 110° and 90°, find the fourth angle. 2 marks

Stuck? EXACTLY one pair parallel = trapezium (NSW definition).

1.5 — Diamond pattern on a jersey. A diamond pattern has 4 equal sides, but the diamond is "tilted" — its angles are NOT right angles.

(a) Name the shape.
(b) Why is it NOT a square?
(c) If one of its angles is 120°, what is the opposite angle (the one across the diagonal)? Explain using what you know about the angle sum. 3 marks

Stuck? 4 equal sides + NOT right angles = rhombus. A rhombus has opposite angles equal — that's a property you'll prove in Lesson 8.

2. Explain your thinking

This question is about communication. Use full sentences. 4 marks

2.1 A classmate looks at a square and writes "It's a square, so it's NOT a rectangle and NOT a rhombus — those are different shapes." In your own words, explain (i) why a square is ALSO a rectangle AND ALSO a rhombus, (ii) how the family tree from Card 5 works, and (iii) why "rectangle" and "rhombus" are still valid (less specific) names for a square.

Stuck? A square HAS the rectangle property (4 right angles) AND the rhombus property (4 equal sides). Both names apply.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Bathroom tile (4 equal sides + 4 right angles)

(a) Most specific name: square.
(b) Also a rectangle (4 right angles), a rhombus (4 equal sides), a parallelogram (inherited from both), and a quadrilateral.

1.2 — Picture frame (20, 30, 20, 30 with right angles)

(a) Not a square because the four sides are NOT all equal (two are 20 cm, two are 30 cm). A square needs all four sides equal.
(b) Most specific name: rectangle.
(c) Also a parallelogram (and a quadrilateral).

1.3 — Kite, three angles 100°, 100°, 60°

(a) Shape: kite.
(b) Fourth angle = 360 − 100 − 100 − 60 = 100° (∠ sum of quad).
Check: 100 + 100 + 60 + 100 = 360° ✓

1.4 — Trapezium sign, three angles 70°, 110°, 90°

(a) Shape: trapezium (NSW definition: exactly one pair of parallel sides).
(b) Fourth angle = 360 − 70 − 110 − 90 = 90° (∠ sum of quad).
Check: 70 + 110 + 90 + 90 = 360° ✓

1.5 — Tilted diamond (4 equal sides, NOT right angles)

(a) Shape: rhombus.
(b) Not a square because a square needs 4 right angles too — this rhombus is tilted, so angles are NOT 90°.
(c) In a rhombus, opposite angles are equal. The two angles "next to" the 120° must each take half of the remaining 360 − 2(120) = 120°, so each is 60°. Then the angle opposite the 120° is also 120°. (Pattern: opposite angles equal; adjacent angles add to 180°.)

2.1 — Explain your thinking (sample response)

(i) A square is BOTH a rectangle AND a rhombus because it satisfies BOTH definitions at once. A rectangle is defined as a quadrilateral with 4 right angles — a square has those. A rhombus is defined as a quadrilateral with 4 equal sides — a square has those too. So calling a square a rectangle, or a rhombus, is not wrong, just less specific.
(ii) The family tree from Card 5 puts the most general shape (quadrilateral) at the top and the most specific (square) at the bottom. A child shape inherits all the properties of every ancestor. A square inherits from rectangle, from rhombus, and from parallelogram, then from quadrilateral.
(iii) "Rectangle" and "rhombus" are still valid names for a square because a square HAS the rectangle property and HAS the rhombus property. We just usually use "square" because it tells the marker the most.

Marking: 1 for square ⊂ rectangle; 1 for square ⊂ rhombus; 1 for naming the hierarchy / family tree; 1 for clear full-sentence explanation.