Parallelogram Theorem
Let’s understand the theorem on parallelogram and how to prove it.
Theorem: Parallelograms on the same base and between the same parallels are equal in area.
To Prove: Area of parallelogram ABCD = Area of parallelogram ABEF
Proof: Let’s assume two parallelograms ABCD and ABEF with the same base DC and between the same parallel lines AB and FC
In the figure given below, the two parallelograms, ABCD and ABEF, lie between the same parallel lines and have the same base. Area ABDE is common between both parallelograms.
Taking a closer look at the two triangles, △BCD and △AEF might be congruent.
Parallelogram Theorem
BC = AE (Opposite sides of a parallelogram),
∠BCD = ∠AEF (These are corresponding angles because BC || AE and CE are the transversal).
∠BDC = ∠AFE (These are corresponding angles because BD || AF and FD are the transversals).
Thus, by the ASA criterion of congruent triangles. These two triangles are congruent, and they must have equal areas.
area(BCD) = area(AEF)
area(BCD) + area(ABDE) = area(AEF) + area(ABDE)
area(ABCD) = area(ABEF)
Hence, parallelograms lying between the same parallel lines and having a common base have equal areas.
Introduction to Parallelogram: Properties, Types, and Theorem
Parallelogram is a two-dimensional geometrical shape whose opposite sides are equal in length and parallel. The opposite angles of a parallelogram are equal in measure.
In this article, we will learn about the definition of a parallelogram, its properties, types, theorem and formulas on the area and perimeter of a parallelogram in detail.
Table of Content
- Parallelogram Definition
- Shape of Parellelogram
- Angles of Parallelogram
- Properties of Parallelogram
- Types of Parallelogram
- Parallelogram Formulas
- Area of Parallelogram
- Perimeter of Parallelogram
- Parallelogram Theorem
- Difference Between Parallelogram and Rectangle
- Solved Examples on Parallelogram
- Real-Life Examples of a Parallelogram
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