Parallelograms with the same base and height have equal area. The triangles on the same base with equal areas lie between the same parallels. In this article we will derive the Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area. We hope that this step-by-step explanation of the below theorem will be helpful for you to score better marks in the exams.
A Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area
Prove the Theorem Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area?
Given
Two parallelograms ABCD and EFCD that have the same base CD and lie between the same parallel AF and CD
To prove that:
ar(ABCD) = ar(EFCD)
Proof:
Since opposite sides of a parallelogram are parallel
AD // BC
With transversal AB
∠DAB =∠CBF (corresponding angles)…. equation 1
ED // FC
With transversal EF
∠DEA =∠CFE (corresponding angles)… equation 2
Also
AD = BC (opposite sides of Parallelogram are equal)…… equation 3
In ∆AED = ∆BFC
∠DEA =∠CFE…….from equation 2
∠DAB =∠CBF…….from equation 1
AD = BC……….. form equation 3
Therefore ∆AED ≅ ∆BFC (AAS congruency)
∆AED ≅ ∆BFC
Hence ar(ABCD) = ar(∆BFC)
Now
ar(ABCD) = ar(∆ADE) + ar(EBCD)
= ar(∆BFC) + ar(EBCD)
= ar(∆EFCD)
Hence proved.
Do Refer:
- Area of a Parallelogram is Equal to that of a Rectangle Between the Same Parallel Lines
- The Area of a Rhombus is Equal to Half the Product of its Diagonals
FAQs on Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area
1. What is the ratio of areas of a triangle and parallelogram so formed between same base and parallel lines?
A triangle and a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the parallelogram. Hence the ratio of the area of the triangle to the area of a parallelogram is 1: 2.
2. Can a triangle and a parallelogram have the same area?
A triangle and a parallelogram have the same base and the same area.
3. What parallelograms have the same area?
Parallelograms that have the same base and the same height will have the same area, the product of the base and height will be equal.