Parallelograms On The Same (Equal) Base and Between The Same Parallels. Now we are going to discuss about the are of two parallelograms which lie on the same or equal base and between the same parallels.
Theorem 1 Parallelograms on the same base and between the same parallels are equal in area.
Given Two parallelograms ABCD and ABEF, which have same basae AB and which are between the same parallel lines AB and FC.
To prove Proof As ABDC is a parallelogram AB = DC ................(1) [ Opposite sides of a parallelogram] Similarly in Parallelogram ABEF AB = EF ................(2) From Eq. (1) and (2) We get DC = EF ...................(3) Subtracting FD from both sides in Eq (3) We get DC- FD = EF - FD CF = DE ..............(4) In AC = BD [ Opposite side of the parallelogram ABDC] AF = BE [ Opposite side of the parallelogram ABEF] CF = DE [ From Equation(4)] So, [ By SSS criterion of congruence ] ...(5) [ Because area of congruent figures is equal] Now, Adding ob both sides of Eq. (5) we get
Hence Proved |
Illustration: In the given figure, . If the ar (ABCD) = 25sq.m find the ar(DEFH)
Solution: In the given figure, we have AB || DC and AD || BC, ABCD is a parallelogram. Again,AD || BC AD || GE and AF || DE ADEG is a parallelogram. Now ||gm ABCD and ||gm ADEG are on the same base AD and between the same parallels AD || BE ar(|| gm ADEG )= ar(|| gm ABCD ) = 25 sq.m. Again, AF || DE and EF || DC DEFH is a parallelogram. Now ||gm DEFH and ||gm ADEG are on the same base DE and between the same parallels DE || AF. ar(|| gm DEFH )= ar(|| gm ADEG ) = 25 sq.m. Hence ar(|| gm DEFH )= 25 sq. m. |
In the given figure we have Two parallelogram and If Area of triangle = 35 sq. cm. Then find out the Area of Parallelogram. | |||
Right Option : D | |||
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Right Option : C | |||||
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Right Option : D | |||||
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