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.

Build · I Do / We Do / You Do

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.

Stuck? Revisit lesson § "The Quadrilateral Family Tree" — square sits at the bottom (most specific), quadrilateral at the top (most general).

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 + _______ = _______ ° ✓

Stuck? The total is fixed at 360°. The fourth angle is whatever closes the sum.

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

Stuck on 3.8? Four equal angles = four right angles. That's enough for one of the special quadrilaterals — but does it pin down whether all sides are equal?

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

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.