Mathematics: Class 8 (Part 1) — Chapter 4: Quadrilaterals

4.1 Introduction to Quadrilaterals

A quadrilateral is a four-sided closed geometric figure. The word is derived from the Latin words quadri (meaning four) and latus (referring to sides). The angles of a quadrilateral are the angles formed between its adjacent sides.

4.2 Rectangles and Squares

Rectangles

Definition: A quadrilateral in which all angles are right angles (90°).
Property 1: Opposite sides are equal in length.
Property 2: Opposite sides are parallel to each other.
Property 3: Diagonals are of equal length ($AC = BD$).
Property 4: Diagonals bisect each other (they intersect at their midpoints).

Squares

Definition: A special type of rectangle in which all four sides are of equal length.
Property 1: All sides are equal and opposite sides are parallel.
Property 2: All angles are equal to 90°.
Property 3: Diagonals are equal in length and bisect each other at right angles (90°).
Property 4: Diagonals bisect the interior angles of the square (dividing them into 45° halves).

Venn Diagram Relationship

Every square is a rectangle, but not every rectangle is a square. The set of all squares is a subset of the set of all rectangles.

4.3 Angles in a Quadrilateral

Angle Sum Property: The sum of all interior angles in any quadrilateral is always 360°.
Proof Idea: Any quadrilateral can be divided into two triangles by drawing a diagonal. Since the sum of angles in a triangle is 180°, the sum for the quadrilateral is $180° + 180° = 360°$.

4.4 Parallelograms

Definition: A quadrilateral in which both pairs of opposite sides are parallel.
Property 1: Opposite sides are equal in length ($AB = CD$ and $AD = BC$).
Property 2: Opposite angles are equal ($\angle A = \angle C$ and tasting $\angle B = \angle D$).
Property 3: Adjacent angles are supplementary (add up to 180°).
Property 4: Diagonals bisect each other (but are not necessarily equal in length).

4.5 Rhombus

Definition: A quadrilateral in which all four sides have the same length.
Property 1: It is a special type of parallelogram, so all properties of a parallelogram apply.
Property 2: Diagonals bisect each other at right angles (90°).
Property 3: Diagonals bisect the interior angles of the rhombus.

4.6 Kite and Trapezium

Kite

Definition: A quadrilateral with two distinct pairs of equal adjacent sides ($AB = BC$ and $CD = DA$).
Property 1: Diagonals are perpendicular to each other.
Property 2: One diagonal bisects the other diagonal.
Property 3: One diagonal bisects the interior angles at its vertices.

Trapezium

Definition: A quadrilateral with at least one pair of parallel opposite sides.
Isosceles Trapezium: A trapezium in which the non-parallel sides are equal in length. In an isosceles trapezium, the base angles are equal.

Chapter Summary & Hierarchy