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When the area of two regions are equal, the

sign ‘=’ is used between them. For example,

in the figure the area of the rectangular

region ABCD = Area of the triangular region

AED.

It is noted that if 'ABCand 'DEFare

congruent, we write 'ABC#'DEF. In this

case, the area of the triangular region ABC =

area of the triangular region DEF.

But, two triangles are not necessarily

congruent when they have equal areas. For

example, in the figure, area of 'ABC = area

of 'DBC but 'ABC and 'DBC are not

congruent.

Theorem 1
Areas of all the triangular regions having same base and lying between the same
pair of parallel lines are equal to one another.

Let the triangular regions ABC and DBC stand on the same base BC and lie between
the pair of parallel lines BCandAD. It is required to prove that, ǻ region ABC = ǻ
region DBC.
Construction : At the points B and C of the line segment BC,draw perpendiculars
BE and CFrespectively. They intersect the line AD or AD produced at the points E
andF respectively. As a result a rectangular region EBCF is formed.
Proof : According to the construction, EBCF is a rectangular region. The triangular
region ABC and rectangular region EBCF stand on the same base BC and lie between

the two parallel line segments BCandED. Hence, ' region ABC =
2

1

(rectangular

region EBCF)

Similarly, ' region DBC =
2

1

(rectangular region EBCF)

?ǻ region ABC = ǻ region DBC (proved).