Introduction to Quadrilaterals

What is a Quadrilateral?

A quadrilateral is a closed, flat (two-dimensional) figure formed by joining exactly four points — no three of which are in a straight line — with four line segments called sides. The word comes from the Latin quadri (four) + latus (side). In the CBSE, Telangana, and Andhra Pradesh Class 8 syllabi, understanding quadrilaterals is the foundation for the entire chapter on their construction using a ruler and compass.

Parts of a Quadrilateral

Consider quadrilateral ABCD. Every quadrilateral has exactly six measurable parts:

• Four Vertices — the corner points A, B, C, D.
• Four Sides — line segments AB, BC, CD, and DA.
• Four Angles — ∠A, ∠B, ∠C, and ∠D, one at each vertex.
• Two Diagonals — line segments AC and BD connecting opposite vertices. They meet at point O inside the figure.

This angle-sum property — that the four interior angles always add up to 360° (four right angles) — is one of the most important facts you will use when constructing quadrilaterals.

Convex and Concave Quadrilaterals

Before learning to construct quadrilaterals, you need to know which type this chapter focuses on. All quadrilaterals fall into one of two broad categories based on how their interior looks.

Types of Quadrilaterals

Quadrilaterals are further classified by the number of parallel sides they have, the equality of their sides, and their angles. The chart below shows how each type is related to the others — from the most general (Trapezium) to the most specific (Square).

Trapezium

A trapezium has exactly one pair of opposite sides that are parallel. In quadrilateral ABCD, if AB ∥ DC, then ABCD is a trapezium. The parallel sides (AB and DC) are called bases, and the other two sides (AD and BC) are called legs or non-parallel sides.

• Parallel sides: AB ∥ DC (one pair only).
• Co-interior angles: Angles on the same non-parallel side add up to 180°. So ∠A + ∠D = 180° and ∠B + ∠C = 180°.

Parallelogram

When both pairs of opposite sides are parallel, the quadrilateral becomes a parallelogram. This unlocks a rich set of properties that are used constantly in board exam problems.

• Parallel sides: AB ∥ DC and AD ∥ BC.
• Equal opposite sides: AB = DC and AD = BC.
• Equal opposite angles: ∠A = ∠C and ∠B = ∠D.
• Supplementary adjacent angles: ∠A + ∠B = 180°, ∠B + ∠C = 180°, etc.
• Bisecting diagonals: The diagonals cut each other in half — OA = OC and OB = OD.

Rectangle

A rectangle is a parallelogram in which at least one angle is exactly 90°. Because the angles of any parallelogram come in supplementary pairs, as soon as one angle is 90°, all four must be 90°.

• All properties of a parallelogram apply.
• All angles are 90°: ∠A = ∠B = ∠C = ∠D = 90°.
• Equal diagonals: AC = BD (diagonals have the same length — unlike a general parallelogram).
• Diagonals still bisect each other: OA = OC and OB = OD.

Rhombus

A rhombus is a parallelogram in which all four sides are equal in length. You can think of it as a "pushed-over" square. Its most distinctive extra property is that its diagonals cross at right angles.

• All properties of a parallelogram apply.
• All four sides are equal: AB = BC = CD = DA.
• Diagonals bisect at right angles: ∠AOB = 90° — this is the key extra property of a rhombus.
• Opposite angles are equal; adjacent angles are supplementary.

Square

A square is the most symmetric quadrilateral. It satisfies both the condition of a rectangle (all angles = 90°) and the condition of a rhombus (all sides equal). It can be defined as a rectangle with two adjacent sides equal, or as a rhombus with one right angle.

• All properties of a rectangle AND a rhombus apply.
• All sides equal: AB = BC = CD = DA.
• All angles 90°: ∠A = ∠B = ∠C = ∠D = 90°.
• Equal diagonals that bisect each other at right angles — both the rectangle property and the rhombus property together.

Quick Comparison: All Types at a Glance

The table below summarises the key properties of each type. Use this as a revision reference before exams — questions in Telangana and AP board exams frequently ask you to identify a quadrilateral from its given properties.

Common Mistakes to Avoid

• Forgetting the angle sum: Many students try to construct a quadrilateral without checking that all four angles add up to 360°. Always verify this first.
• Confusing rhombus and square: A rhombus has all sides equal but its angles need not be 90°. A square is a rhombus with right angles.
• Treating a rectangle as always having equal diagonals in parallelograms: Equal diagonals are a special property of rectangles (and squares), not of all parallelograms.
• Assuming a quadrilateral with equal sides is a square: Equal sides alone only guarantee a rhombus. You also need right angles for it to be a square.
• Using the wrong number of measurements: To uniquely construct a quadrilateral, you need five independent measurements (sides + angles + diagonals). Using fewer leads to multiple possible shapes.

What This Introduction Prepares You For

A solid understanding of quadrilateral types and their properties is the entry point to every exercise in this chapter. In the upcoming exercises you will learn to construct quadrilaterals when given different combinations of five measurements — for example, four sides and one diagonal, three sides and two diagonals, or three sides and two included angles.

The properties from this introduction — especially that diagonals of a parallelogram bisect each other and that a square's diagonals are equal and perpendicular — are exactly the clues you use to choose your construction strategy in each exercise.

These concepts also connect directly to other chapters: the angle-sum property links to algebraic expressions when angles are given as expressions, and properties of similar figures connect ahead to triangles in Class 9 and coordinate geometry in Class 10.