Unit: Understanding Quadrilateral
Chapter: Trapezium & Parallelogram
Reference: – What is a Quadrilateral, Trapezium (Trapezoid) Definition, Types of Trapezium (Isosceles, Right, Scalene), Properties of a Trapezium, Parallelogram Definition, Properties of a Parallelogram, Special Parallelograms (Rectangle, Rhombus, Square), Difference Between Trapezium and Parallelogram, Area of Trapezium, Area of Parallelogram, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is a Trapezium (Trapezoid)
- What is a Parallelogram
- Properties of a Trapezium and a Parallelogram
- Difference Between Trapezium and Parallelogram
- Special Types of Parallelograms (Rectangle, Rhombus, Square)
Introduction to Trapezium & Parallelogram
Definition
A quadrilateral is a polygon with four sides, four vertices, and four angles. The sum of interior angles of any quadrilateral is 360°. The two special types of quadrilaterals we study in this chapter are the trapezium (called trapezoid in the US) and the parallelogram.
When we study trapezium and parallelogram, we essentially ask:
"How many pairs of parallel sides does this quadrilateral have?"
The answer determines whether it is a trapezium (one pair) or a parallelogram (two pairs).
Importance
- Used in architecture, engineering, and design (roofs, bridges, beams)
- Foundation for understanding area and perimeter calculations
- Helps develop spatial reasoning and geometric intuition
- Appears in real-world objects (tables, windows, frames)
Example
A trapezium (trapezoid) has exactly one pair of parallel sides, like a tabletop with one wider end. A parallelogram has two pairs of parallel sides, like a leaning bookcase or a diamond shape.
Subtopics
1. Trapezium (Trapezoid)
Definition: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The non-parallel sides are called legs.
US Terminology: Trapezoid (US) = Trapezium (UK/India)
Types of Trapezium:
Isosceles Trapezium: The legs are equal in length. Base angles are equal. Diagonals are equal.
Right Trapezium: One leg is perpendicular to the bases (has two right angles).
Scalene Trapezium: All sides are of different lengths.
Properties of a Trapezium:
- Exactly one pair of opposite sides is parallel
- The angles adjacent to each leg are supplementary (sum to 180°)
- The median (segment joining midpoints of legs) equals half the sum of the bases
Area of a Trapezium: Area = (1/2) × (sum of parallel sides) × (height)
Formula: A = (1/2) × (a + b) × h
Where a and b are the lengths of the parallel sides, and h is the perpendicular distance between them.
Example – Area of Trapezium: Bases = 8 cm and 12 cm, height = 5 cm
Area = (1/2) × (8 + 12) × 5 = (1/2) × 20 × 5 = 10 × 5 = 50 cm²
2. Parallelogram
Definition: A quadrilateral with both pairs of opposite sides parallel.
Properties of a Parallelogram:
- Opposite sides are parallel
- Opposite sides are equal in length
- Opposite angles are equal
- Adjacent angles are supplementary (sum to 180°)
- Diagonals bisect each other (cut each other in half)
- Each diagonal divides the parallelogram into two congruent triangles
Area of a Parallelogram: Area = base × height
Formula: A = b × h
Where b is the length of any side, and h is the perpendicular distance to the opposite side (height).
Example – Area of Parallelogram: Base = 10 cm, height = 6 cm
Area = 10 × 6 = 60 cm²
3. Special Types of Parallelograms
Rectangle: A parallelogram with all angles 90° (right angles).
Properties: All properties of parallelogram + diagonals are equal
Rhombus: A parallelogram with all sides equal.
Properties: All properties of parallelogram + diagonals are perpendicular bisectors of each other + diagonals bisect the interior angles
Square: A parallelogram with all sides equal AND all angles 90°.
Properties: All properties of rectangle + all properties of rhombus (diagonals equal, perpendicular, bisect angles)
4. Venn Diagram Relationship
Square ⊂ Rhombus ⊂ Parallelogram
Square ⊂ Rectangle ⊂ Parallelogram
Rectangle and Rhombus are both Parallelograms, but their intersection is Square
Solved Examples
Example 1 – Area of Trapezium: Find the area of a trapezium with parallel sides 15 cm and 25 cm and height 8 cm.
Solution: A = (1/2) × (15 + 25) × 8 = (1/2) × 40 × 8 = 20 × 8 = 160 cm²
Answer: 160 cm²
Example 2 – Area of Parallelogram: Find the area of a parallelogram with base 12 m and height 5 m.
Solution: A = 12 × 5 = 60 m²
Answer: 60 m²
Example 3 – Parallelogram Angle: In parallelogram ABCD, angle A = 70°. Find angle B.
Solution: Adjacent angles in a parallelogram are supplementary.
Angle A + angle B = 180° → 70° + angle B = 180° → angle B = 110°
Answer: 110°
Example 4 – Isosceles Trapezium: In an isosceles trapezium, the legs are 6 cm each, bases are 10 cm and 14 cm. Find the perimeter.
Solution: Perimeter = sum of all sides = 10 + 14 + 6 + 6 = 36 cm
Answer: 36 cm
Common Mistakes to Avoid
Mistake 1 – Confusing trapezium with parallelogram
Thinking a trapezium has two pairs of parallel sides is wrong.
Correct understanding: Trapezium has exactly one pair; parallelogram has two pairs.
Mistake 2 – Using the wrong height for area
Height must be perpendicular distance between bases (trapezium) or between base and opposite side (parallelogram).
Correct understanding: Height is NOT the length of the slanted side.
Mistake 3 – Assuming all parallelograms are rectangles
A parallelogram can have slanted sides. Only rectangles have 90° angles.
Correct understanding: Parallelogram = opposite sides parallel; rectangle = parallelogram with right angles.
Mistake 4 – Forgetting that a square is both a rectangle and a rhombus
A square satisfies all properties of both rectangles and rhombuses.
Correct understanding: Square ⊂ Rectangle and Square ⊂ Rhombus.
Mistake 5 – Thinking all trapeziums are isosceles
Only isosceles trapeziums have equal legs and equal base angles.
Correct understanding: Trapeziums can be scalene or right-angled too.
Mistake 6 – Misidentifying the median formula for trapezium
The median (mid-segment) = (a + b)/2, not (a × b)/2.
Correct understanding: Median is the average of the two bases.
Quick Reference Summary
Trapezium (Trapezoid): Exactly 1 pair of parallel sides
Parallelogram: 2 pairs of parallel sides
Trapezium Area: A = (1/2) × (a + b) × h (a, b = parallel sides, h = height)
Parallelogram Area: A = b × h (b = base, h = height)
Rectangle: Parallelogram with all angles 90°
Rhombus: Parallelogram with all sides equal
Square: Parallelogram with all sides equal AND all angles 90°
Properties Shared: Opposite sides parallel (parallelogram family), diagonals bisect each other (parallelogram family)
Key Difference: Trapezium has 1 pair of parallel sides; Parallelogram has 2 pairs