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Answers
Step-by-step explanation:
Consider two parallelograms ABCD and ABEF, on the same base AB, and between the same parallels, as shown below:
Parallelograms - Same base same parallel
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:
Parallelogram and Rectangle - Same base same parallel
Clearly, their areas will be equal. Now, the length and height (width) of the rectangle have been marked as l and w respectively. Therefore,
area(ABCD) = area(ABEF) = l × w
This means that the area of any parallelogram is equal to the product of its base and its height (the height of a parallelogram can be defined as the distance between its base and the opposite parallel).
Now, consider the following figure, which shows a parallelogram ABCD and a triangle ABE on the same base AB and between the same parallels:
Parallelogram and Triangle - Same base same parallel
What will be the relation between the areas of these two figures? Let us complete the parallelogram ABEF, as shown below:
Parallelogram and Triangles - Same base same parallel
Now, we have:
area(
Δ
A
B
E
) = ½ area(ABFE)
= ½ area(ABCD)
Thus, the area of the triangle is exactly half of the area of the parallelogram. Let us define the height of a triangle as the distance between the base and the parallel through the opposite vertex. We can therefore say that the area of the triangle will be:
Area = ½ × base × height
This is shown below:
Area of a triangle
Note that any of the three sides of the triangle can be taken as the base, but then the height will change accordingly.
Download SOLVED Practice Questions of Parallelograms - Same Base, Same Parallels for FREE
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Answer:
Given =》 Two llgm ABCD and ABEF On The same base AB and between the same parallel AB and FC
To prove=》 ar(llgm ABCD ) = ar(llgmABEF )
proof=》In △ADF and △BCF , we have
AD= BC ( opp. sides of a llgm)
LADF= LBCE( corresponding)
DF= CE( DC= FE ( each equal to AB )=》DC-FC=FE-FC=DF=CE)
△ADF congruent △BCE(by SAS)
And So , ar( △ADF) = ar( △BCE)
Congruent have equal area