prove palleogram are base between same parallels are equal in area
Answers
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Answer:
what will be the relation between the areas of these two parallelograms?
theorem: parallelograms on the same base and between the same parallels are equal in area.
proof: consider the figure presented above. can you see that
δ
b
c
e
and
δ
a
d
f
will be congruent? this is easy to show. we have:
bc = ad (opposite sides of a parallelogram are equal)
∠
b
c
e
=
∠
a
d
f
(corresponding angles)
∠
b
e
c
=
∠
a
f
d
(corresponding angles)
by the asa criterion, the two triangles are congruent, which means that their areas are equal. now,
area(abcd) = area(abed) + area(
δ
b
c
e
)
similarly,
area(abef) = area(abed) + area(
δ
a
d
f
)
clearly,
area(abcd) = area(abef)
this completes the proof.
next, consider a parallelogram abcd and a rectangle abef on the same base and between the same parallels: