Mathematics • Year 7 • Unit 3 • Lesson 7
Introducing Quadrilaterals — Mixed Challenge
Mix naming, the 360° angle sum, the family-tree hierarchy, and the trickiest definitions (kite vs rhombus, NSW trapezium). Spot a common naming error, then design a quadrilateral that fits multiple categories.
1. Mixed problems
Decide whether each question wants a NAME (use the most specific name) or a NUMBER (use the 360° rule). Show working where there's an equation. 2 marks each
1.1 Three angles of a quadrilateral are 85°, 95° and 110°. Find the fourth.
1.2 A quadrilateral has 2 pairs of parallel sides but its angles are NOT right angles. Name it using its most specific name.
1.3 A quadrilateral has angles 90°, 90°, 90° and (180 − 90)°. Find the fourth angle and name the shape.
1.4 A quadrilateral has 4 equal sides AND 4 equal angles. Name it and explain why no other answer is possible.
1.5 A quadrilateral has angles 3x°, 4x°, 5x° and 6x°. Find x, then state each angle.
1.6 A kite has angles 70°, x°, 110° and x° (with the two equal angles between the unequal-length sides). Find x.
2. Find the mistake
A Year 7 student tried to name a quadrilateral with 4 equal sides but NOT right angles. Their working is shown. Exactly one line is wrong. Spot it, explain the slip, then write the corrected reasoning. 3 marks
Student's working — name a quadrilateral with 4 equal sides and NO right angles:
Line 1: 4 equal sides means it's a square.
Line 2: But the question says NO right angles.
Line 3: So it's a "tilted square" — which is just a square that's been rotated.
Line 4: Final answer: square.
(a) Which line contains the conceptual error?
(b) Explain in one or two sentences why "tilted square" isn't right here.
(c) Write out the correct naming.
Stuck? Rotating a square doesn't change its angles. A real square ALWAYS has 4 right angles, no matter the orientation.3. Open-ended challenge — design a quadrilateral with multiple names
This question has more than one correct answer. 4 marks
3.1 Describe a quadrilateral that you can correctly name in at least THREE different ways using the special-quadrilateral family tree. For each name, give the property that lets you use it.
Requirements: (i) list the side and angle properties of your shape; (ii) check those properties satisfy the 360° angle sum; (iii) list every category from the family tree it belongs to; (iv) justify each name in one short sentence.
Bonus: Find a quadrilateral that can be named in exactly TWO ways (no more) — explain why no third name fits.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 85° + 95° + 110° + ? = 360°
Sum = 290. Fourth = 360 − 290 = 70° (∠ sum of quad).
1.2 — 2 pairs parallel, NOT right angles
Most specific name: parallelogram (if 4 sides were also equal it would be a rhombus; if it had right angles it would be a rectangle).
1.3 — 90°, 90°, 90°, (180 − 90)°
Fourth = 90°. Check: 90 + 90 + 90 + 90 = 360° ✓ (∠ sum of quad). Shape: at least a rectangle (could be a square if its sides happened to all be equal).
1.4 — 4 equal sides AND 4 equal angles
4 equal angles → each = 360 ÷ 4 = 90° → 4 right angles. Combined with 4 equal sides, this gives a square. No other answer is possible because the square is defined by exactly these two conditions.
1.5 — 3x + 4x + 5x + 6x = 360
18x = 360, x = 20. Angles = 3(20), 4(20), 5(20), 6(20) = 60°, 80°, 100°, 120°. Check: 60 + 80 + 100 + 120 = 360° ✓.
1.6 — Kite: 70 + x + 110 + x = 360
2x + 180 = 360, so 2x = 180 and x = 90°. The kite has angles 70°, 90°, 110°, 90°. Check: 70 + 90 + 110 + 90 = 360° ✓.
2 — Find the mistake
(a) The error is on Line 1.
(b) "4 equal sides" alone is NOT the definition of a square. A square needs 4 equal sides AND 4 right angles. With 4 equal sides but no right angles, the shape is a rhombus. "Tilted square" is also wrong — rotating a square doesn't remove its right angles.
(c) Correct naming:
Line 1 (fixed): 4 equal sides → either rhombus or square, depending on the angles.
Line 2: Question says NO right angles.
Line 3: Without right angles, it's NOT a square.
Line 4 (fixed): Final answer = rhombus.
3 — Open-ended challenge (sample solution)
Sample design. A 4 cm × 4 cm tile with all corners 90°. Properties: 4 equal sides (4 cm each), 4 right angles (90° each), 2 pairs of parallel sides.
Angle check: 90 + 90 + 90 + 90 = 360° ✓ (∠ sum of quad).
Family tree categories — FIVE names:
1. Square — because 4 equal sides AND 4 right angles.
2. Rectangle — because 4 right angles.
3. Rhombus — because 4 equal sides.
4. Parallelogram — because 2 pairs of parallel sides (inherited from being a rectangle or rhombus).
5. Quadrilateral — because 4 sides.
Bonus (exactly 2 names). A rectangle with sides 3 cm, 5 cm, 3 cm, 5 cm and four right angles. Names: rectangle (4 right angles) and parallelogram (2 pairs parallel, inherited). It is NOT a square (sides not all equal), NOT a rhombus (sides not all equal), NOT a trapezium (BOTH pairs are parallel, not just one), NOT a kite (no pair of adjacent equal sides forming the kite shape — opposite sides are equal, not adjacent).
Marking: 1 for clearly listing properties; 1 for the 360° check; 1 for at least 3 distinct names; 1 for the bonus or for justifying each name in a short sentence.