Mathematics • Year 7 • Unit 3 • Lesson 7
Introducing Quadrilaterals
Build fluency with the six special quadrilaterals: square, rectangle, parallelogram, rhombus, trapezium, kite. Every quadrilateral has angle sum 360°. Always choose the MOST specific name a shape's properties allow.
1. I do, fully worked example
Read every line. Each step shows you how to walk the family tree and pick the most specific name.
Problem. A quadrilateral has all four sides equal in length, but its angles are NOT right angles. What is its most specific name?
Step 1, List what we KNOW.
4 equal sides ✓ | 4 right angles ✗
Reason: "all four sides equal" is the defining feature of a rhombus.
Step 2, Walk the family tree.
4 equal sides AND 4 right angles → square. We DON'T have right angles, so NOT a square.
Reason: a square needs BOTH conditions; missing one of them rules it out.
Step 3, Identify the most specific name.
A quadrilateral with 4 equal sides (but no right angles) is a rhombus.
It's also a parallelogram, but "rhombus" is more specific, always pick the most specific name.
Answer: Rhombus.
2. We do, fill in the missing steps
Fill in each blank. The problem uses the 360° angle sum to find a missing angle. 4 marks
Problem. Three angles of a quadrilateral are 80°, 100° and 120°. Find the fourth angle.
Step 1, Recall the rule.
Interior angles of a quadrilateral sum to _______ ° (∠ sum of quad)
Step 2, Add the three known angles.
80 + 100 + 120 = _______ °
Step 3, Subtract from 360°.
Fourth angle = 360 − _______ = _______ °
Step 4, Check.
80 + 100 + 120 + _______ = _______ ° ✓
3. You do, independent practice
Mix of "name the quadrilateral" and "find the missing angle". Show working where there's an equation. Always pick the MOST specific name.
Foundation, naming & recall
3.1 A quadrilateral has 4 right angles. Its sides are NOT all equal. Name it. 1 mark
3.2 A quadrilateral has 2 pairs of adjacent equal sides but no parallel sides. Name it. 1 mark
3.3 State the angle sum of any quadrilateral. 1 mark
3.4 Give the NSW definition of a trapezium. 1 mark
Standard, apply the 360° rule
3.5 Three angles of a quadrilateral are 90°, 90° and 90°. Find the fourth and name the shape. 2 marks
3.6 Three angles of a quadrilateral are 70°, 130° and 95°. Find the fourth. 2 marks
Extension, multiple names from the family tree
3.7 A quadrilateral has 4 equal sides AND 4 right angles. List every category from the family tree it belongs to. 3 marks
3.8 A quadrilateral has all four angles equal in size. (i) What is each angle? (ii) Can you say it is definitely a square? Explain. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (80° + 100° + 120° + ? = 360°)
Step 1: sum = 360° (∠ sum of quad).
Step 2: 80 + 100 + 120 = 300°.
Step 3: Fourth angle = 360 − 300 = 60°.
Step 4: 80 + 100 + 120 + 60 = 360° ✓
3.1-4 right angles, sides NOT all equal
Rectangle. (If all four sides were equal too, it would be a square.)
3.2-2 pairs of adjacent equal sides, no parallel sides
Kite.
3.3, Angle sum of any quadrilateral
360°. (Reason: split the shape with a diagonal into two triangles, each 180°.)
3.4, NSW trapezium definition
A trapezium has exactly one pair of parallel sides (not "at least one", if both pairs are parallel, it's a parallelogram).
3.5-90° + 90° + 90° + ? = 360°
Fourth angle = 360 − 270 = 90° (∠ sum of quad). Four right angles → it's at least a rectangle (and a square if the sides also happen to all be equal).
3.6-70° + 130° + 95° + ? = 360°
Sum of three = 295. Fourth = 360 − 295 = 65° (∠ sum of quad).
3.7-4 equal sides + 4 right angles
Most specific name: square. Walking up the family tree, the shape is correctly called: square, rectangle (4 right angles), rhombus (4 equal sides), parallelogram (2 pairs of parallel sides, inherited from rectangle or rhombus), and of course quadrilateral (4 sides).
3.8, All four angles equal
(i) 4 equal angles summing to 360° → each = 360 ÷ 4 = 90° (∠ sum of quad).
(ii) NOT definitely a square. Four right angles means it's at least a rectangle. It's only a square if the four sides also happen to be equal, but the question only tells us about angles, so we can't conclude that. A rectangle with two long sides and two short sides also has four equal (right) angles.