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# Triangle and Parallelogram of Same Base and Between the Same Parallels

#### Summary

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The parallelograms are on the same base and between the same parallels.
Area of a Parallelogram = b × h
In both parallelograms:
•    The bases are equal.
•    The heights are equal.
Area of parallelogram ABCD = Area of parallelogram ABGH.
Parallelograms on the same base and between the same parallels are equal in area.
Theorem:
Prove that the area of the triangle is equal to half the area of the parallelogram, if they are on the same base and are between the same parallels.

Given:
Δ ABP and  parallelogram ABCD are on same base AB and between the same parallels AB and PC.
To Prove: ar (ΔABP) =1/2 ar ( ||gm ABCD)
Construction: Draw BQ || AP
Proof:
ar ( ||gm ABQP) = ar ( || gm ABCD (Parallelograms on the same base and between the same parallels are equal in area.)
Δ ABP ≅ Δ BQP
ar (Δ ABP) = ar (Δ BQP)
ar (Δ ABP) =  (½) ar (|| gm ABQP)
ar (Δ ABP) =  (½) ar (|| gm ABCD).

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