Question

prove that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a quadrilateral is a parallelogram.

Answer 1

hello ☺


To prove :-

PS || QR  and SR || PQ.

Quadrilateral PQRS is a parallelogram


Proof:-

Draw diagonal BD.As PS is the midsegment of ▲ ABD, we can say that PS || BD.

QR is the midsegment of ▲ BCD,

so QR || BD.

∵ PS || BD and QR || BD
so PS || QR.

SR is the midsegment of ▲ ACD

so now SR || AC.As PQ is the midsegment of ▲ ABC

PQ || AC.∵ SR || AC and PQ || AC by transitivity

SR || PQ.∵ PS || QR and SR || PQ,

∴ quadrilateral PQRS is a parallelogram

hence proved..


thank you

Answer 2

To prove that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a quadrilateral is a parallelogram:-

parallelogram is a quadrilateral with both pairs of opposite sides parallel proved by these theorems:-

1. (THEOREM: If a quadrilateral has diagonals which bisect each other, then it is a parallelogram).
2. (THEOREM: If a quadrilateral has one set of opposite sides which are both congruent and parallel, then it is a parallelogram).

$\color{red}{</strong><strong>Two</strong><strong>}$$\color{red}{</strong><strong>colomn</strong><strong>}$$\color{red}{</strong><strong>Proof</strong><strong>:</strong><strong>-</strong><strong>}$

We can use one of these ways in a two-column proof. Bear in mind that, to challenge you, most problems involving parallelograms and proofs will not give you all the information about the presented shape. Here, for example, you are given a quadrilateral and told that its opposite sides are congruent.

We can use one of these ways in a two-column proof. Bear in mind that, to challenge you, most problems involving parallelograms and proofs will not give you all the information about the presented shape. Here, for example, you are given a quadrilateral and told that its opposite sides are congruent.[insert drawing of quadrilateral GOAT with sides GO ≅ TA and TG ≅ OA]

We can use one of these ways in a two-column proof. Bear in mind that, to challenge you, most problems involving parallelograms and proofs will not give you all the information about the presented shape. Here, for example, you are given a quadrilateral and told that its opposite sides are congruent.[insert drawing of quadrilateral GOAT with sides GO ≅ TA and TG ≅ OA]Statement Reason:

1. GO ≅ TA and TG ≅ OA (Given)
2. Construct segment TO Construct a diagonal
3. TO ≅ TO Reflexive Property
4. △GOT ≅ △ TOA Side-Side-Side Postulate: If three sides of one △
5. are congruent to three sides of another △, then the two △ are congruent
6. ∠GTO ≅ ∠ TOA CPCTC: Corresponding parts of congruent △ are
7. ∠GOT ≅ ∠ OTA congruent
8. GO ∥ TA and TG ∥ OA Converse of the Alternate Interior Angles

Theorem: If a transversal cuts across two lines and the alternate interior angles are congruent, then the lines are parallel

GOAT Definition of a parallelogram: A quadrilateral

with two pairs of opposite sides parallel

The two-column proof proved the quadrilateral is a parallelogram by proving opposite sides were parallel.

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