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**(1) Prove that a diagonal of a parallelogram divides it into two triangles of equal area.
**

In ABD and CDB, we have

And,

So, by SSS criterion of congruence, we have

Hence,

**(2) Prove that parallelograms on the same base and between the same parallels are equal in area.
**

And, Â Â Â Â Â Â Â Â [ â¸ª and ]

So, by SAS criterion of congruence, we have

Â Â â€¦..(i)

Now,

Â Â [Using(i)]

Hence,

**(3) Prove that the area of a parallelogram is the product of its base and the corresponding altitude.
**

Â Â Â [By rect. Area axiom area of a rectangle = Base X Height]

Hence,

**(4) Prove that parallelograms on equal bases and between the same parallels are equal in area.
**

Now,

Â [ and ]

**(5) Prove that triangles on the same bases and between the same parallels are equal in area.
**

And,

is a parallelogram.Similarly, is a parallelogram.

Now, parallelograms ECQP and BCAD are on the same base BC, and between the same parallels.

Â Â Â â€¦.(i)

We know that the diagonals of a parallelogram divides it into two triangles of equal area.

Â â€¦..(ii)

And, Â ....(iii)

Now, [ From (i)]

Â Â Â [From (ii) and (iii)]

Hence,

**(6) Prove that the area of a triangle is half the product of any of its sides and the corresponding altitude.
**

And,

BCDA is a parallelogram.

Since AC is a diagonal of parallelogram BCDA.

[BC is the base and AL is the corresponding altitude of parallelogram BCDA]

**(7) Prove that if a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to the half of the parallelogram.
**

Â Â Â â€¦..(i)

Now,

Â Â [ (from (i)]

**(8) Prove that the area of a trapezium is half the product of its height and the sum of parallel sides.
**

Â [ and ]

**(9) Prove that triangles having equal areas and having one side of one of the triangles, equal to one side of the other, have their corresponding altitudes equal.
**

CN and RT are the altitudes corresponding to AB and PQ respectively of the two triangles.

Â Â Â â€¦.(i)

Similarly, we have,

Â Â â€¦..(ii)

Now,

Â Â [ (Given)]

**(10) Prove that if each diagonal of a quadrilateral separates it into two triangles of equal area, then the quadrilateral is a parallelogram.
**

and

Â Â Â â€¦..(i)

But,

Â Â Â [Using (i)]

â€¦.(ii)

Since diagonal BD of the quadrilateral ABCD separates it into triangles of equal area.

Â Â â€¦.(iii)

But,

Â Â Â [Using(iii)]

Â Â â€¦..(iv)

From (ii) and (iv), we get

Since ABC and ABD are on the same base AB. Therefore they must have equal corresponding altitudes.

i.e. Altitude from C of = Altitude from D of

Similarly,

Hence, quadrilateral ABCD is a parallelogram.

**(11) Prove that the area of a rhombus is half the product of the lengths of its diagonals.
**

and

ar(rhombus)

ar(rhombus)

ar(rhombus)

ar(rhombus)

**(12) Prove that diagonals of a parallelogram divide it into four triangles of equal area.
**

and

Also, the median of a triangle divides it into two equal parts.

Now, in , BO is the median.

Â â€¦.(i)

In , CO is the median

Â Â Â â€¦..(ii)

In , DO is the median

Â Â â€¦.(iii)

From (i), (ii) and (iii), we get

**(13) Prove that if the diagonals AC and BD of a quadrilateral ABCD, intersect at O and separate the quadrilateral into four triangles of equal area, then the quadrilateral ABCD is parallelogram.
**

Thus, ABD and ABC have the same base AB and have equal areas. So, their corresponding altitudes must be equal.

Altitude from Altitude from C of

Similarly, we have, .

Hence, quadrilateral ABCD is a parallelogram.

**(14) Prove that a median of a triangle divides it into two triangles of equal area.
**

Â [ Multiplying both sides by AL]

ALITER Since ABD and ADC have equal bases and the same altitude AL.

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very helpful and good for studies

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