**Rhombuses Rectangles and Squares Worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on proving rhombuses, rectangles and squares.

Before look at the worksheet, if you would like to know the stuff related to rhombuses, rectangles and squares,

**Question 1 :**

Decide whether the statement given below is always, sometimes or never true.

"A rhombus is a rectangle"

**Question 2 :**

Decide whether the statement given below is always, sometimes or never true.

"A parallelogram is a rectangle"

**Question 3 :**

ABCD is a rectangle. What else do you know about ABCD ?

**Question 4 : **

In the diagram at the right, PQRS is a rhombus. What is the value of y ?

**Question 5 : **

Prove that the diagonals of a rhombus are perpendicular.

**Question 1 :**

Decide whether the statement given below is always, sometimes or never true.

"A rhombus is a rectangle"

**Answer :**

The statement is sometimes true.

In the Venn diagram above, the regions for rhombuses and rectangles overlap. If the rhombus is a square, it is a rectangle.

**Question 2 :**

Decide whether the statement given below is always, sometimes or never true.

"A parallelogram is a rectangle"

**Answer :**

The statement is sometimes true. Some parallelograms are rectangles.

In the Venn diagram above, we can see that some of the shapes in the parallelogram box are in the region for rectangles, but many aren’t.

**Question 3 :**

ABCD is a rectangle. What else do you know about ABCD ?

**Answer : **

Because ABCD is a rectangle, it has four right angles by the definition. The definition also states that rectangles are parallelograms.

So, ABCD has all the properties of a parallelogram :

• Opposite sides are parallel and congruent.

• Opposite angles are congruent and consecutive angles are supplementary.

• Diagonals bisect each other.

**Question 4 : **

In the diagram at the right, PQRS is a rhombus. What is the value of y ?

**Answer : **

All four sides of a rhombus are congruent, so we have

RS = PS

Substitute RS = 5y - 6 and PS = 2y + 3.

5y - 6 = 2y + 3

Add 6 to both sides.

5y = 2y + 9

Subtract 2y from

3y = 9

Divide both sides by 3.

y = 3

**Question 5 : **

Prove that the diagonals of a rhombus are perpendicular.

**Answer : **

Let us consider the rhombus PQRS as shown below.

In the rhombus PQRS above, we have to prove

PR ⊥ QS

PQRS is a rhombus, so we have

PQ ≅ QR

Because the rhombus PQRS is also a parallelogram, its diagonals bisect each other.

So, we have

PX ≅ RX

PQ ≅ QR

Using SSS postulate, we have

ΔPXQ ≅ ΔRXQ

So, we have

∠PXQ ≅ ∠RXQ ----> ∠PXQ = ∠RXQ

Because ∠PXQ, ∠RXQ form a linear pair and ∠PXQ = ∠RXQ, we have

∠PXQ = ∠RXQ = 90°

Hence, we have

PR ⊥ QS

After having gone through the stuff given above, we hope that the students would have understood, "Rhombuses rectangles and squares worksheet".

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