AREAS OF PARALLELOGRAMS AND

TRIANGLES

Group : 2Class : IX - D

PARALLELOGRAM • A quadrilateral is called a parallelogram, if both

pairs of its opposite sides are parallel. In the adjoining figure, ABCD is a quadrilateral in which  AB   DC and AD   BC.  ∥ ∥So, ABCD is a parallelogram.

• PRORETIES OF PARALLELOGRAM.• Opposite sides are parallel and congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• The diagonals bisect each other.

TRIANGLEA triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted  ABC.

•The part of the plane enclosed by a simple closed The part of the plane enclosed by a simple closed figure is called a planar region corresponding to that figure is called a planar region corresponding to that figure. The magnitude or measure of this planar region figure. The magnitude or measure of this planar region is called its area. This magnitude or measure is always is called its area. This magnitude or measure is always expressed with the help of a number (in some unit) expressed with the help of a number (in some unit) such as 5 cmsuch as 5 cm22 ,8m ,8m22 ,3 hectares , etc. so, we can say ,3 hectares , etc. so, we can say that area of a figure is a number (in some unit) that area of a figure is a number (in some unit) associated with the part of the plane enclosed by the associated with the part of the plane enclosed by the figure with the following two properties: figure with the following two properties: •(1) If A and B are two congruent figures, then ar(A) = (1) If A and B are two congruent figures, then ar(A) = ar(B)ar(B)•(2) If a planar region formed by a figure T is made up (2) If a planar region formed by a figure T is made up of two non-overlapping planar regions formed by of two non-overlapping planar regions formed by figures P and Q, then ar(T)= ar(P) + ar(Q)figures P and Q, then ar(T)= ar(P) + ar(Q)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.•If two figures A and B are congruent, they must have equal areas. However, the converse of this statement is not true.•Two figures having equal areas need not be congruent.

COGRUENCY

Figures on the Same Base and Between the Same

Parallels

Parallelograms on the same base and between the same parallels.

Two triangles on the same base (or equal bases) in between the same parallels are equal in area.

Two triangles having the same

base (or equal bases) and equal areas lie between the same parallel