# Some Special Parallelograms

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## Rhombus

A rhombus is **a quadrilateral with sides of equal length**. Since the opposite sides of a rhombus have the same length, it is **also a type of parallelogram**.

An interesting property of a rhombus is that it is a **special case** of a kite and parallelogram. This is despite the fact that the properties of a kite and a parallelogram have nothing in common. Therefore, a **rhombus consists of properties of a parallelogram and a kite as well.**

As for the diagonals of a rhombus:

**Property:** **The diagonals of a rhombus are perpendicular bisectors of one another.**

- A rhombus is also a
where we know, the diagonals each other. - Therefore, considering the triangles- ∆AOB and ∆COB, we get OA =
- We also have AB =
as they are of a rhombus and OB is a side. - Thus, by
congruency criterion- ∆ AOB ≅ ∆ - We also see that ∠AOB and ∠COB are equal and have an angle sum of
° - Thus, m ∠AOB = m ∠COB =
° - Therefore, the diagonals of a rhombus perpendicularly bisect each other.

Lets's try out an example.

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**Example 7: RICE is a rhombus. Find x, y, z. Justify your findings.**

**Solution:**

x = OE = OI (diagonals

y = OR = OC =

z = side of the rhombus =

**Thus, x = 5 , y = 12 and z = 13**

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## Rectangle

A rectangle is **a type of parallelogram in which every angle equal**. Therefore, it has all the properties of a parallelogram:

(1) Opposite sides of equal length and

(2) Diagonals bisect each other.

Since all the angles are equal in a rectangle and the total sum of the interior angles for a quadrilateral is 360°, we get that:

**Each angle = 90°**

**Note:** In a parallelogram, the diagonals can be of different lengths but in a rectangle, the diagonals are of equal lengths.

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**Property:** **The diagonals of a rectangle are of equal length**.

- Considering the triangles ∆ ABC and ∆BAD: AB =
as it is a side. - We also have: BC = AD as they are
sides and m ∠A = m ∠B = ° as all angles are angles. - Therefore, by
criterion: ∆ ABC ≅ ∆ - Which gives us, AC =
- Therefore, in a rectangle the diagonals are equal in length and also bisect each other.

From the above conclusion, we can say that: **In a rectangle the diagonals, besides being equal in length bisect each other.**

Let's try out an example.

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**Example 8: RENT is a rectangle. Its diagonals meet at O. Find x, if OR = 2x + 4 and OT = 3x + 1**

**Solution:**

OT is half of the diagonal TE and OR is half of the diagonal RN (as diagonals

Diagonals are

So, their halves are also equal.

Therefore, 3x + 1 = 2x + 4 ⇒ x =

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## Square

A square is further a special case of a

Therefore, like earlier, **a square has properties of a rectangle while having all sides equal**.

In a square, the diagonals have the following properties:

(i) bisect one another (Square is a type of parallelogram)

(ii) are of equal length (Square is a type of rectangle) and

(iii) bisect each other perpendicularly.

So, now we need to prove the following property:

**Property:** **The diagonals of a square are perpendicular bisectors of each other.**

- With O as the point of intersection for the diagonals, consider the triangles- ∆AOD and ∆COD
- We have: AD =
as all sides are equal, OD is a side and OA = as diagonals each other. - Therefore, by
congruency criterion: ∆AOD ≅ ∆ - Thus we get, m∠AOD = m∠
- Since the angles are a linear pair we get, m∠AOD = m∠COD =
° - Therefore, the diagonals of a square are perpendicular bisectors of each other.

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## Let's Solve

**A mason has made a concrete slab. He needs it to be rectangular. Answer Yes/No for the following below mentioned properties that need to be vertified in order to make sure that it is rectangular?**

We need to check for all the properties of a rectangle which include:

(1) Diagonals are equal in length.

(2) All angles have a measure of 90°.

(3) Opposite sides are parallel and equal

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**Answer the following:**

(1) Both rhombus and square have all sides of equal length. In a square, the additional requirement is of having all right angles. Since, the total sum of the interior angles of a quadrilateral is 360°, we get that: 4 x Each angle = 360° .Therefore, Each angle = 90°

(2) No, a trapezium cannot have equal angles. A trapezium is a quadrilateral with exactly one pair of parallel sides. If the angles were equal, it would imply that the non-parallel sides are also equal in length, making the shape a parallelogram, not a trapezium.

(3) All angles of a trapezium need not be equal, only atleast one pair of opposite sides needs to be parallel.