Quadrilaterals

• Properties of parallelograms, rectangles, rhombuses, squares, and trapeziums.
• How opposite sides, angles, and diagonals behave in each type.
• The mid-point theorem and its use in proving line segments parallel or equal.
• Angle-sum property: the four interior angles of any quadrilateral sum to 360°.
• Real applications in architecture, tiling, and land surveying.

1. Parallelogram — Opposite sides parallel and equal. Opposite angles equal. Diagonals bisect each other. Adjacent angles supplementary.

2. Rectangle — Parallelogram with all angles 90°. Diagonals equal in length and bisect each other.

3. Rhombus — Parallelogram with all sides equal. Diagonals perpendicular and bisect each other; they bisect vertex angles.

4. Square — Both rectangle and rhombus: equal sides, 90° angles, equal and perpendicular diagonals.

5. Trapezium — Exactly one pair of opposite sides parallel. The parallel sides are called bases.

6. Mid-point theorem — The segment joining mid-points of two sides of a triangle is parallel to the third side and half its length.

7. Where it shows up — Floor tiles (squares/rectangles), kite shapes, bridge trusses, and parallelogram-shaped fields in agriculture.

In parallelogram ABCD, ∠A = (3x + 10)° and ∠B = (2x + 20)°. Find all angles.

Step 1 — Adjacent angles in a parallelogram are supplementary
Step 2 — (3x + 10) + (2x + 20) = 180
Step 3 — 5x + 30 = 180  →  5x = 150  →  x = 30
Step 4 — ∠A = 3(30) + 10 = 100°,  ∠B = 80°
Step 5 — Opposite angles equal: ∠C = 100°,  ∠D = 80°
Answer: 100°, 80°, 100°, 80°

Application: A rectangular park 80 m × 50 m has diagonal paths. Both diagonals are equal (√(80² + 50²) ≈ 94.3 m), so either diagonal gives the same shortcut distance.

• Assuming every quadrilateral is a parallelogram.
• Forgetting diagonals of a rhombus are perpendicular but not necessarily equal.
• Using angle-sum 360° only for rectangles — it applies to all quadrilaterals.
• Confusing trapezium (one pair parallel) with parallelogram (two pairs parallel).

• Name a quadrilateral that is both a rhombus and a rectangle.
• If one angle of a parallelogram is 65°, find the other three.
• State one property unique to a rhombus (not true for every parallelogram).
• In a trapezium, two angles on the same leg are 110° and 70° — verify they are co-interior.

Open the Practice tab for graded questions on Quadrilaterals.