Question

Explain it in detail:- Theorem 1: Parallelograms on the same base and between the same parallel sides are equal in area.​

Answer 1

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Proof: Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Proof:

Consider the figure given below:

Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth.

Therefore, area of parallelogram ABCD = AB x AL

Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.

In ∆ADF and ∆BCE,

AD=BC (∴ABCD is a parallelogram ∴ AD=BC)

AF=BE (∴ABEF is a parallelogram ∴AF=BE)

∠ADF=∠BCE (Corresponding Angles)

∠AFD=∠BEC (Corresponding Angles)

∠DAF =∠CBE (Angle Sum Property)

∆ADE ≅ ∆BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD)=Area(ABED) + Area(BCE)

Area(ABCD)=Area(ABED)+Area(ADF)

Area(ABCD)=Area(ABEF)

Hence, the area of parallelograms on the same base and between the same parallel sides is equal.

Answer 2

Answer:

Here is your answer:-

Step-by-step explanation:

Proof: Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Proof:

Consider the figure given below:

Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth.

Therefore, area of parallelogram ABCD = AB x AL

Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.

In ∆ADF and ∆BCE,

AD=BC (∴ABCD is a parallelogram ∴ AD=BC)

AF=BE (∴ABEF is a parallelogram ∴AF=BE)

∠ADF=∠BCE (Corresponding Angles)

∠AFD=∠BEC (Corresponding Angles)

∠DAF =∠CBE (Angle Sum Property)

∆ADE ≅ ∆BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD)=Area(ABED) + Area(BCE)

Area(ABCD)=Area(ABED)+Area(ADF)

Area(ABCD)=Area(ABEF)

Hence, the area of parallelograms on the same base and between the same parallel sides is equal.