Theorem 1:
Parallelograms on the same base and between
the same parallels are equal in area.

Given:
and are two
parallelograms.
To prove:
Proof :
In and
(Corresponding angles, ,and AF transversal.)
(Corresponding
angles, , and transversal.)
So,
(Third angle
of the triangles, by angle sum property)
(Opposite sides of parallelogram )
(ASA
rule)
So
(From
the fig.)
.
Theorem 2: Area of triangle is half of parallelogram
with common base and between same parallel.
Given:
Let
and
parallelogram be with
common base .

To prove :
Construction: Draw , .
Extend Up to .
Proof:
AS ,
(Distance
between two parallel lines is equal)
Now,
.
Theorem
3: Two triangles on the same base (or
equal bases) and between the same parallels are equal in area.
Given:
And be two triangles
with common base and between same
Parallel lines

To
prove:
Construction: Draw , and ,
Let line meets at and at .
Proof:
(Triangle and parallel
gram with common base and
Between parallel lines) …(1)
And …(2)
is parallelogram
is parallelogram.
Both the parallelogram have same base and
between two parallel lines,
….(3)
So,
(from and
)
Theorem 4 : Two triangles having the same base and equal
areas lie between the same
Parallels.
Given:
and have equal
base BC.
,
Let
,

To prove:
Triangles lie between two parallel
lines.
i.e.
Proof :
As
AS
corresponding heights are equal, we say point and lie on
the line parallel
to , through .
Hence
the result.