A quadrilateral is said to be a parallelogram if each pair of its opposite sides are parallel. Since the sides of the parallelogram are parallel, the rules for parallel lines and transversals are applicable to the parallelogram also.
Property 1: Opposide sides of a parallelogram are of equal lengths, i.e., AB = DC, AD = BC
Property 2: Opposite angles of a parallelogram are of equal measures, i.e., ∠A = ∠C, ∠B = ∠D
Property 3: Adjacent angles of a parallelogram are supplementary, i.e., ∠A + ∠D = 180°, ∠C + ∠B = 180°, ∠A + ∠B = 180°, ∠D + ∠C = 180°
Property 4: Each diagonal of a parallelogram divides it into two congruent triangles, i.e., \(\bigtriangleup ABC \cong \bigtriangleup ADC\)
Property 5: The diagonals of a parallelogram bisect each other at O, i.e., AO = OC, OD = OB
Theorem 1: If a pair of opposite sides of a quadrilateral are equal in length and parallel, it is a parallelogram.
Theorem 2: In a parallelogram, opposite sides and opposite angles are equal.
Theorem 1: Parallelograms on the same base and between the same parallel lines are equal in area.
Area[Parallelogram ABCD] = Area[Parallelogram ABEF]
Theorem 2: If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to one-half that of the parallelogram.
Area of △ ABE = \(\frac{1}{2}\)[Area of Parallelogram ABCD]
Theorem 3: The area of a parallelogram is the product of any of its sides and the corresponding altitude.
Area of Parallelogram ABCD = AB × AE
All the above properties of the parallelogram are valid for rectangle, square and rhombus. Following are the specific properties for square, rectangle and rhombus:
Rectangle: The diagonals of a rectangle are equal.
Square: The diagonals of square are equal and cut each other at right angles.
Rhombus: (i) The angles of a rhombus are bisected by the diagonals. (ii) The diagonals of a rhombus cut at right angles.
