Given that RSTU is a parallelogram and ST is congruent to TU, explain why RSTU must also be a rhombus.

Given:

RSTU is a parallelogram and ST is congruent to TU.

To find:

Why RSTU must also be a rhombus?

Solution:

We know that the opposite sides of a parallelogram are parallel and congruent.

RSTU is a parallelogram. So,

[tex]RS\parallel TU[/tex] and [tex]ST\parallel RU[/tex]            [Property of parallelogram]            ...(i)    

[tex]RS\cong TU[/tex] and [tex]ST\cong RU[/tex]            [Property of parallelogram]            ...(ii)    

It is given that,

[tex]ST\cong TU[/tex]                ...(iii)

From (ii) and (iii), we get

[tex]RS\cong TU\cong ST\cong RU[/tex]          ...(iv)

From (i) and (iv), we get RSTU is a rhombus because all sides are congruent and appositive sides are parallel.

A parallelogram with each side equal is known as rhombus.

As for the given parallelogram all the sides are equal, therefore it should be rhombus.

Parallelogram is a closed shaped quadrilateral polygon in which opposite sides are equal and parallel and opposite angles are equal.

Rhombus is a closed shaped quadrilateral polygon in which all the sides are equal and opposites sides are parallel also opposite angles are equal.

Given information-

In the given problem [tex]RSTU[/tex] is a parallelogram.

In the given parallelogram [tex]ST[/tex] is congruent to [tex]TU[/tex].

The image of parallelogram is attached below.

For a parallelogram opposite sides are equal. thus,

[tex]RS=TU\\RU=ST[/tex]

As In the given parallelogram [tex]ST[/tex] is congruent to [tex]TU[/tex]. Thus,

[tex]ST=TU[/tex]

Now the side [tex]ST[/tex] is equal to the side [tex]RU[/tex] and side [tex]RS[/tex] is equal to the side [tex]TU[/tex]. Thus,

[tex]RS=ST=TU=RU[/tex]

All the sides of a rhombus is equal in length. A parallelogram with each side equal is known as rhombus.

As all the sides are equal of the given parallelogram. Thus the given parallelogram is rhombus.

Hence, as for the given parallelogram all the sides are equal, therefore it should be rhombus.

Learn more about the parallelogram and rhombus here;

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