Mathematics • Year 7 • Unit 3 • Lesson 9
Rhombuses, Squares, Kites and Trapeziums
Build fluency with the remaining special quadrilaterals: rhombus (4 equal sides, diagonals perpendicular), square (inherits from rectangle AND rhombus), kite (2 pairs of adjacent equal sides, one pair of equal angles), and trapezium (exactly one pair of parallel sides).
1. I do — fully worked example
Read every line. The working uses the fact that a rhombus is also a parallelogram, so all parallelogram properties (opposite angles equal, adjacent angles co-interior) apply.
Problem. Rhombus PQRS has ∠P = 110°. Find the other three angles.
Step 1 — Opposite angles equal (rhombus is a parallelogram).
∠R = ∠P = 110° (opp. angles of parallelogram)
Reason: a rhombus is a parallelogram, so opposite angles are equal.
Step 2 — Adjacent angles co-interior (180°).
∠Q = 180 − 110 = 70° (co-int. angles, PQ ∥ SR)
Step 3 — Last angle is opposite ∠Q.
∠S = ∠Q = 70° (opp. angles)
Check: 110 + 70 + 110 + 70 = 360° ✓.
Answer: ∠Q = 70°, ∠R = 110°, ∠S = 70°.
2. We do — fill in the missing steps
Fill in each blank. This problem uses the kite property: ONE pair of opposite angles is equal. 4 marks
Problem. Kite ABCD has AB = AD and CB = CD. ∠A = 90° and ∠C = 30°. Find ∠B and ∠D.
Step 1 — Use angle sum of the quadrilateral.
∠A + ∠B + ∠C + ∠D = _______ ° (∠ sum of quad)
Step 2 — Use the kite property: ∠B = ∠D.
90 + ∠B + 30 + ∠B = 360 ⇒ 2∠B + _______ = 360
Step 3 — Solve.
2∠B = 360 − _______ = _______
∠B = _______ °, so ∠D = _______ °.
Step 4 — Check.
90 + _______ + 30 + _______ = 360° ✓
3. You do — independent practice
Use the property that fits each shape and state the reason in brackets.
Foundation — single-property recall
3.1 Rhombus ABCD has ∠A = 80°. Find ∠B. 1 mark
3.2 A square has diagonals meeting at 90°. What angle does each diagonal make with each side of the square? 1 mark
3.3 Trapezium ABCD has AB ∥ DC and ∠A = 95°. Find ∠D. 1 mark
3.4 State the defining feature of a rhombus and the defining feature of a kite. How are they different? 1 mark
Standard — combine two properties
3.5 Kite ABCD has ∠A = 100°, ∠C = 60° (the symmetry axis goes through AC). Find ∠B and ∠D. 2 marks
3.6 Trapezium ABCD has AB ∥ DC. ∠A = 115° and ∠B = 75°. Find ∠C and ∠D. 2 marks
Extension — apply the family tree
3.7 Rhombus PQRS has ∠P = 60°. Find every other angle. Then state every category from the family tree this rhombus belongs to. 2 marks
3.8 A quadrilateral has 4 equal sides AND its diagonals are equal in length. Show that it must be a square. (Hint: 4 equal sides → rhombus; equal diagonals → also a rectangle.) 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (kite ABCD, ∠A = 90°, ∠C = 30°)
Step 1: sum = 360° (∠ sum of quad).
Step 2: 2∠B + 120 = 360.
Step 3: 2∠B = 360 − 120 = 240. ∠B = 120°, so ∠D = 120°.
Step 4: 90 + 120 + 30 + 120 = 360° ✓
3.1 — Rhombus, ∠A = 80°
Co-interior with ∠A: ∠B = 180 − 80 = 100° (co-int. angles, AB ∥ DC).
3.2 — Square diagonals
Each diagonal bisects a 90° corner of the square, so the angle between a diagonal and a side = 90 ÷ 2 = 45°.
3.3 — Trapezium ABCD, AB ∥ DC, ∠A = 95°
∠A and ∠D are co-interior on leg AD: ∠D = 180 − 95 = 85° (co-int. angles, AB ∥ DC).
3.4 — Rhombus vs kite
Rhombus: 4 equal sides (all four the same length). Kite: 2 pairs of ADJACENT equal sides (the two pairs have DIFFERENT lengths from each other). Difference: a rhombus is "4 equal", a kite is "2 + 2 of different lengths". Every rhombus is technically also a kite, but not vice versa.
3.5 — Kite ABCD, ∠A = 100°, ∠C = 60°
∠B = ∠D (kite property — pair of angles BETWEEN unequal sides). Sum: 100 + ∠B + 60 + ∠B = 360, so 2∠B = 200 and ∠B = 100°. So ∠D = 100°.
Check: 100 + 100 + 60 + 100 = 360° ✓.
3.6 — Trapezium, AB ∥ DC, ∠A = 115°, ∠B = 75°
∠D = 180 − 115 = 65° (co-int. angles on leg AD).
∠C = 180 − 75 = 105° (co-int. angles on leg BC).
Check: 115 + 75 + 105 + 65 = 360° ✓.
3.7 — Rhombus PQRS, ∠P = 60°
∠R = 60° (opp. angles); ∠Q = 180 − 60 = 120° (co-int.); ∠S = 120° (opp. angles). Check: 60 + 120 + 60 + 120 = 360° ✓.
Family tree categories: rhombus (4 equal sides), parallelogram (rhombus ⊂ parallelogram), quadrilateral. NOT a rectangle (no right angles) and NOT a square.
3.8 — 4 equal sides AND equal diagonals → square
4 equal sides → the shape is a rhombus. Equal diagonals → the shape is a rectangle (only a rectangle has equal diagonals among parallelograms). Rhombus AND rectangle → 4 equal sides AND 4 right angles → square. Therefore the quadrilateral is a square.