G4: Quadrilaterals & Triangles
Apply the properties and definitions of quadrilaterals and triangles
📋 Key Concepts
Triangles: 3-sided polygons with angles summing to 180°.
Quadrilaterals: 4-sided polygons with angles summing to 360°.
📝 Types of Triangles
| Type | Sides | Angles | Key Properties |
|---|---|---|---|
| Equilateral | All equal | All 60° | 3 lines of symmetry |
| Isosceles | Two equal | Two equal | 1 line of symmetry |
| Scalene | All different | All different | No symmetry |
| Right-angled | Any | One 90° | Pythagoras applies |
Example 1
In an isosceles triangle, the equal angles are 68°. Find the third angle.
Solution:
68° + 68° + x = 180°
136° + x = 180°
x = 44°
Example 2
Find the angles in an equilateral triangle.
Solution: All angles are equal. 180° ÷ 3 = 60° each.
📝 Types of Quadrilaterals
Square
- All sides equal
- All angles 90°
- 4 lines of symmetry
- Diagonals bisect at 90°
- Diagonals are equal in length
Rectangle
- Opposite sides equal and parallel
- All angles 90°
- 2 lines of symmetry
- Diagonals are equal in length
Parallelogram
- Opposite sides equal and parallel
- Opposite angles equal
- No lines of symmetry
- Diagonals bisect each other
Rhombus
- All sides equal
- Opposite angles equal
- 2 lines of symmetry
- Diagonals bisect at 90°
Trapezium
- One pair of parallel sides
- Co-interior angles sum to 180°
Isosceles Trapezium
- One pair of parallel sides
- Non-parallel sides equal
- Base angles equal
- 1 line of symmetry
Kite
- Two pairs of adjacent equal sides
- One pair of equal angles
- 1 line of symmetry
- Diagonals intersect at 90°
📝 Comparing Quadrilaterals
| Property | Square | Rectangle | Rhombus | Parallelogram |
|---|---|---|---|---|
| All sides equal | ✓ | ✗ | ✓ | ✗ |
| All angles 90° | ✓ | ✓ | ✗ | ✗ |
| Diagonals equal | ✓ | ✓ | ✗ | ✗ |
| Diagonals at 90° | ✓ | ✗ | ✓ | ✗ |
Example 3
A parallelogram has angles of 112° and x. Find x.
Solution:
Opposite angles are equal, so one angle is 112°.
Co-interior angles: 112° + x = 180°
x = 68°
Angles are: 112°, 68°, 112°, 68°
🧮 Worked Examples — Exam Style
Angle questions on quadrilaterals lean on two facts: angles in a triangle sum to 180°, angles in a quadrilateral to 360° — plus the special properties of each shape.
Two angles of a triangle are 55° and 65°. Find the third angle.
Angles in a triangle sum to 180°.
55 + 65 = 120 → 180 − 120 = 60°
B1 for 60°. If the triangle is isosceles and one base angle is given, remember the OTHER base angle equals it.
A parallelogram has one angle of 70°. Write down the sizes of the other three angles.
Opposite angles are equal: one more angle = 70°.
Co-interior (allied) angles sum to 180°: the other two = 180 − 70 = 110° each.
B1 for 70°, B1 for 110° and 110°. Sketching the parallelogram and labelling is accepted working.
Exam wording to expect: "Work out the size of angle …" (give reasons), "Write down" (properties recall), or symmetry counting.
❓ Practice Questions
Q1: In an isosceles triangle, the angle between the equal sides is 50°. Find the other two angles.
Q2: How many lines of symmetry does a rhombus have?
Q3: In a kite, one diagonal is 8 cm and the other is 12 cm. At what angle do they intersect?
Q4: A trapezium has parallel sides. If one angle is 72°, what is the angle adjacent to it on the same side?
Q5: Name the quadrilateral with all sides equal but not all angles 90°.
Q6: Two angles of a triangle are 55° and 65°. Find the third angle.
Q7: Work out the size of each interior angle of a regular pentagon.
✅ Answers
- (180° - 50°) ÷ 2 = 65° each
- 2 lines of symmetry
- 90° (kite diagonals always intersect at right angles)
- 180° - 72° = 108° (co-interior angles)
- Rhombus
🧮 Worked Examples — Harder Exam Style
In triangle ABC, AB = AC. Angle BAC = 36°. D lies on AC so that BD = BC. Work out angle ABD, giving reasons.
AB = AC → base angles equal: angle ABC = angle ACB = (180 − 36) ÷ 2 = 72°
BD = BC → triangle BCD isosceles → angle BDC = angle BCD = 72°
Angle DBC = 180 − 72 − 72 = 36°
Angle ABD = angle ABC − angle DBC = 72 − 36 = 36°
M1 for base angles 72°, M1 for the isosceles step in BCD, A1 for 36°. This 'silver ratio' chain is a real AQA Foundation 4-marker — every line needs its reason.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Apex 50° between the equal sides → two base angles equal: (180 − 50) ÷ 2 = 65° each
Q2. A rhombus has 2 lines of symmetry (its diagonals)
Q3. Kite diagonals meet at 90° — always, by symmetry
Q4. Co-interior angles between parallel sides: 180 − 72 = 108°
Q5. All sides equal, angles not 90° → rhombus
Q6. 180 − (55 + 65) = 60°
Q7. Sum = (5 − 2) × 180 = 540° → regular → 540 ÷ 5 = 108° each. M1 for 540, A1 for 108.
📌 Key Facts to Memorise
- Triangle angles: 180° · Quadrilateral angles: 360°
- Isosceles: 2 equal sides, 2 equal base angles, 1 line of symmetry
- Equilateral: 3 equal sides, 60° each, 3 lines of symmetry
- Parallelogram: opposite sides parallel and equal; opposite angles equal; co-interior angles sum 180°
- Rhombus: equal sides, 2 symmetry lines (the diagonals), diagonals ⊥
- Kite: 2 pairs of adjacent equal sides; diagonals ⊥; 1 symmetry line
- Trapezium: one pair of parallel sides; co-interior angles on the parallel sides sum to 180°
🗺️ In the Exam
Angle properties: 2–4 marks per paper. "Give a reason for each stage of your working" is the standard instruction — reasons are named facts (base angles in an isosceles triangle are equal), not restated arithmetic.
Links to: angles in parallel lines (G), polygon interior/exterior angles (G), constructions (G).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Learn the properties of each quadrilateral thoroughly
- Remember: a square is also a rectangle, rhombus, and parallelogram
- In isosceles triangles, mark the equal sides and equal angles
- Draw diagrams to help visualise the problem
- Check your answer by adding all angles - they should match the expected total
🧠 Problem-Solving Strategies
Problem-Solving
Learn the key properties of each quadrilateral: parallelogram (opposite sides equal and parallel, opposite angles equal), rhombus (all sides equal, diagonals perpendicular), rectangle (all angles 90°, diagonals equal), kite (two pairs of adjacent sides equal, one pair of opposite angles equal). Use these properties to find missing angles and sides.
Multi-Step Problem
In a parallelogram ABCD, ∠A = (4x − 10)° and ∠B = (2x + 30)°. Find all four angles.
Solution: Adjacent angles in a parallelogram sum to 180°. 4x − 10 + 2x + 30 = 180. 6x + 20 = 180. x = 26⅔. ∠A = 4(26⅔) − 10 = 96⅔°. ∠B = 2(26⅔) + 30 = 83⅓°. ∠C = ∠A = 96⅔°, ∠D = ∠B = 83⅓°. Check: 96⅔ + 83⅓ = 180° ✓
⚠️ Common Errors
Watch Out!
1. Wrong: Saying a rhombus has all angles equal (like a square) Correct: A rhombus has all SIDES equal, but only opposite angles are equal. A square is a special rhombus with all angles 90°.
2. Wrong: Thinking a kite has two pairs of parallel sides Correct: A kite has NO parallel sides. It has two pairs of adjacent (next to each other) equal sides, not opposite equal sides.
3. Wrong: Assuming all quadrilaterals have diagonals that bisect each other Correct: Only parallelograms (including rectangles, rhombuses and squares) have bisecting diagonals. A kite and a general trapezium do not.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: ABCD is a rhombus with diagonals AC = 10 cm and BD = 8 cm. (a) Find the length of each side of the rhombus. (b) Find the area of the rhombus. (c) One angle of the rhombus is θ. Find θ to the nearest degree.
(a) The diagonals of a rhombus bisect each other at right angles. Half-diagonals: 5 cm and 4 cm. Side = √(5² + 4²) = √41 ≈ 6.4 cm.
(b) Area = ½ × d₁ × d₂ = ½ × 10 × 8 = 40 cm².
(c) In the right-angled triangle with legs 5 and 4: tan θ = 5/4 (or 4/5 depending on which angle). θ = tan⁻¹(5/4) ≈ 51° and the other angle = 180 − 51 ≈ 129°. The acute angle is approximately 51° and the obtuse angle is approximately 129°.
Mark scheme: M1 halving diagonals, A1 side ≈ 6.4cm, M1 area formula, A1 40cm², M1 trig, A1 angles
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student says: "All squares are rectangles, and all rectangles are parallelograms, so all squares are parallelograms."
(a) Is the conclusion logically valid?
(b) Give two additional properties a square has that a general parallelogram does not.
(c) Is every rhombus a square? Explain.
Answers: (a) Yes — the logic is valid. Squares have all rectangle properties (90° angles, equal diagonals) and rectangles have all parallelogram properties (opposite sides equal and parallel, diagonals bisect). (b) A square has all sides equal AND all angles 90° AND equal diagonals. A general parallelogram may have unequal sides, non-right angles, and unequal diagonals. (c) No — a rhombus has all sides equal but does NOT need all angles to be 90°. Only a rhombus with 90° angles is a square.
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