Subscribe to our YouTube channel for the latest videos, updates, and tips.
Parallelograms on Same Base and between Same Parallels
Parallelograms on same base and between same parallels have same area.
|
In the adjoining figure, ABCD and BCEF are the two parallelograms on the same base BC and between the parallels BC and AE. |
 |
Therefore, area of parallelogram ABCD = Area of parallelogram BCEF.
Explanation:
Draw a parallelogram ABCD on a thick sheet of paper or a cardboard sheet.
Now, draw a line segment DE as shown in the figure.
Next, cut a triangle A’D’E’ congruent to triangle ADE in a separate sheet with the help of a tracing paper and place ∆ A’D’E’ in such a way that A’D’ coincides with BC as shown in adjoining figure.
Note that there are two parallelograms ABCD and EE’CD on the same base DC and between the same parallels AE’ and DC. What can you say about their areas?
As ∆ADE ≅ ∆ A’ D’ E’
Therefore Area (ADE) = Area (A’ D’ E’)
Also Area (ABCD) = Area (ADE) + Area (EBCD)
= Area (A’D’E’) + Area (EBCD)
= Area (EE’CD)
So, the two parallelograms are equal in area.
Solved Example:
Parallelograms ABCD and ABEF are situated on the opposite sides of AB in such a way that D, A, F are not collinear. Prove that DCEF is a parallelogram, and parallelogram ABCD + parallelogram ABEF = parallelogram DCEF.
Construction: D, F and C, E are joined.
Proof: AB and DC are two opposite sides of parallelogram ABCD,
Therefore, AB ∥ DC and AB = DC
Again, AB and EF are two opposite sides of parallelogram ABEF
Therefore, AB ∥ EF and AB ∥ EF
Therefore, DC ∥ EF and DC = EF
Therefore, DCEF is a parallelogram.
Therefore, ∆ADF and ∆BCE, we get
AD = BC (opposite sides of parallelogram ABCD)
AF = BE (opposite sides of parallelogram ABEF)
And DF = CE (opposite sides of parallelogram CDEF)
Therefore, ∆ADF ≅ ∆BCE (side – side – side)
Therefore, ∆ADF = ∆BCE
Therefore, polygon AFECD - ∆BCE = polygon AFCED - ∆ADF
Parallelogram ABCD + Parallelogram ABEF = Parallelogram DCEF
Figure on Same Base and between Same Parallels
Parallelograms on Same Base and between Same Parallels
Parallelograms and Rectangles on Same Base and between Same Parallels
Triangle and Parallelogram on Same Base and between Same Parallels
Triangle on Same Base and between Same Parallels
8th Grade Math Practice
From Parallelograms on Same Base and between Same Parallels to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
Worksheet on Average | Word Problem on Average | Questions on Average
May 17, 25 05:37 PM
In worksheet on average interest we will solve 10 different types of question. Find the average of first 10 prime numbers. The average height of a family of five is 150 cm. If the heights of 4 family -
How to Find the Average in Math? | What Does Average Mean? |Definition
May 17, 25 04:04 PM
Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities. -
Problems Based on Average | Word Problems |Calculating Arithmetic Mean
May 17, 25 03:47 PM
Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed. -
Rounding Decimals | How to Round a Decimal? | Rounding off Decimal
May 16, 25 11:13 AM
Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate -
Worksheet on Rounding Off Number | Rounding off Number | Nearest 10
May 15, 25 05:12 PM
In worksheet on rounding off number we will solve 10 different types of problems. 1. Round off to nearest 10 each of the following numbers: (a) 14 (b) 57 (c) 61 (d) 819 (e) 7729 2. Round off to
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.