Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area

Here we will prove that parallelogram on the same base and between the same parallel lines are equal in area.

Given: PQRS and PQMN are two parallelograms on the same base PQ and between same parallel lines PQ and SM.

To prove: ar(parallelogram PQRS) = ar(parallelogram PQMN).

Construction: Produce QP to T.

Proof:

            Statement

           Reason

1. PS = QR.

1. Opposite sides of the parallelogram PQRS.

2. PN = QM.

2. Opposite sides of the parallelogram PQMN.

3. ∠SPT = ∠RQT.

3. Opposite sides PS and QR are parallel and TPQ is a transversal.

4. ∠NPT = ∠MQT.

4. Opposite sides PN and QM are parallel and TPQ is a transversal.

5. ∠NPS = ∠MQR.

5. Subtracting statements 3 and 4.

6. ∆PSN ≅ ∆RQM

6. By SAS axiom of congruency.

7. ar(∆PSN) ≅ ar(∆RQM).

7. By area axiom for congruent figures.

8. ar(∆PSN) + ar(quadrilateral PQRN) = ar(∆RQM) + ar(quadrilateral PQRN)

8. Adding the same area on both sides of the equality in statement 7.

9. ar(parallelogram PQRS) = ar(parallelogram PQMN). (Proved)

9. By addition axiom for area.





9th Grade Math

From Parallelogram on the Same Base and Between the Same Parallel Lines 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.



Share this page: What’s this?

Recent Articles

RSS

  1. Perimeter of a Triangle | Perimeter of a Triangle Formula | Examples

    Apr 25, 24 05:13 PM

    Perimeter of a Triangle
    We will discuss here how to find the perimeter of a triangle. We know perimeter of a triangle is the total length (distance) of the boundary of a triangle. Perimeter of a triangle is the sum of length…

    Read More

  2. Perimeter of a Rectangle | How to Find the Perimeter of a Rectangle?

    Apr 25, 24 03:45 PM

    Perimeter of a Rectangle
    We will discuss here how to find the perimeter of a rectangle. We know perimeter of a rectangle is the total length (distance) of the boundary of a rectangle. ABCD is a rectangle. We know that the opp…

    Read More

  3. Perimeter of a Square | How to Find the Perimeter of Square? |Examples

    Apr 25, 24 12:54 PM

    Perimeter of a Square
    We will discuss here how to find the perimeter of a square. Perimeter of a square is the total length (distance) of the boundary of a square. We know that all the sides of a square are equal. Perimete…

    Read More

  4. Dividing 3-Digit by 1-Digit Number | Long Division |Worksheet Answer

    Apr 24, 24 03:46 PM

    Dividing 3-Digit by 1-Digit Number
    Dividing 3-Digit by 1-Digit Numbers are discussed here step-by-step. How to divide 3-digit numbers by single-digit numbers? Let us follow the examples to learn to divide 3-digit number by one-digit nu…

    Read More

  5. Symmetrical Shapes | One, Two, Three, Four & Many-line Symmetry

    Apr 24, 24 03:45 PM

    Symmetrical Figures
    Symmetrical shapes are discussed here in this topic. Any object or shape which can be cut in two equal halves in such a way that both the parts are exactly the same is called symmetrical. The line whi…

    Read More