###### Rhombus Parallelogram Trapezium Fun

We have to learn the following geometrical shape like Point, Rays, Line, Line segment, Triangle, Square, Rectangle in the previous post. Now we learn the following geometrical shape like Rhombus, Parallelogram, Trapezium etc.

**What is Rhombus?**

Rhombus is just like a modified shape of a square as shown in above figure. It is a closed two dimensional plane figure. It has some specific unique properties. Rhombus is a type of quadrilateral and a special case of a parallelogram. Its diagonals are intersect with each other at right angle same as to the square.

All the sides are equal in length. i.e. AB = BC = CD = DA = 5cm. Just like a square.

There are two diagonals AC and BD and represented by d_{1} and d_{2} respectively. The diagonals bisect each other at right angle to just like a square.

# It forms four right angled triangle with two diagonal and its sides.

# Every opposite triangle are equal in area.

# Its opposite sides are parallel to each other. i.e. AB||DC and AD||BC

A rhombus has four internal angle. The sum of two adjacent angles is 180^{0}.

∠A + ∠B = 180°

∠B + ∠C = 180°

∠C + ∠D = 180°

∠A + ∠D = 180°

As shown in above figure, The opposite angles are equal.

∠A = ∠C

∠B = ∠D

The sum of the four ∠A, ∠B, ∠C and ∠D is equal to 360^{0}.

∠A + ∠B + ∠C + ∠D = 360^{0}.

**Perimeter of Rhombus:**

The perimeter of a rhombus is the sum of all the four equal side length. We can say the total length of its boundaries is called the perimeter of Rhombus. The perimeter is always denoted by the capital letter ‘P’.

The perimeter of Rhombus

= AB + BC + CD + DA

= 4 AB = 4 BC = 4 CD = 4 DA

P = 4 x Side

**Area of Rhombus:**

The Area of Rhombus is the region which is covered by its boundaries on a plane paper. It is two dimensional plane figure. The area of rhombus is the half of the product of its two diagonals. The area is always denoted by the capital letter ‘A’.

Area of Rhombus

= ½ (AC x BD)

= ½ (d_{1} x d_{2}) metre^{2}

**Properties of Rhombus**

- Opposite angle are equal.
- The sum of two adjacent angles is 180
^{0}. - The sum of four angles is 360
^{0}. - Opposite sides are parallel.
- All the sides are equal.
- Diagonals bisect each other at 90
^{0}. - Diagonal bisect the angles also.
- The perimeter of rhombus = 4 x side
- The area of rhombus = ½ (d
_{1}x d_{2})

**What is parallelogram?**

A parallelogram is just like a modified shape of a rectangle as shown in above figure. It is closed two dimensional plane figure. It has also some specific unique properties.

As shown in above figure, A parallelogram is a quadrilateral two dimensional plane figure. It has four sides and four angles.

Its two opposite sides are parallel and equal in length. like AB||DC and AD||BC and AB = DC and AD = BC respectively.

Its opposite angles are also equal like ∠A = ∠C and ∠B = ∠D.

The sum of two adjacent angles is equal to 180^{0}. Like ∠A + ∠D = 180^{0} and ∠B + ∠C = 180^{0}.

The sum of all the interior angles is equal to 360^{0}. Like ∠A + ∠D + ∠B + ∠C = 360^{0}.

As shown in above figure, There are different part of a parallelogram. The blue coloured line is Slant Length. The violet coloured line is the Base. The green coloured line is the Height.

**The perimeter of parallelogram:**

The perimeter of a parallelogram is the sum of the length of the four sides AB, BC, CD and DA. Here, Two opposite sides are equal in length. Like AB = DC and AD = BC.

The perimeter of parallelogram

= AB + BC + CD + DA

= AB + BC + AB + BC

= (AB + AB) +( BC + BC)

= 2 AB + 2 BC

=2 (AB + BC)

= 2 (Length + Width)

**Area of Parallelogram:**

As shown in above figure, The Area of Parallelogram is the region which is covered by its boundaries on a plane paper. It is two dimensional plane figure. The area of parallelogram is the product of its base and height. Here, The base of parallelogram is AB = DC = ‘b’. The height of parallelogram is ‘h’. The area is always denoted by the capital letter ‘A’.

Area of parallelogram

= Base x Height

= b x h metre^{2}

**Angle of parallelogram:**

As shown in above figure, There are four internal angles. Like ∠A, ∠D, ∠B and ∠C.

The opposite angles ∠A and ∠C are equal. Like ∠A = ∠C. The other two opposite angles ∠D and ∠B are equal. Like ∠D = ∠B.

The sum of two adjacent angles is 180^{0}. Like ∠A + ∠B = ∠B + ∠C = ∠C + ∠D = ∠D + ∠A = 180^{0}.

The sum of all the four angles are equal to 360^{0}. Like ∠A + ∠D + ∠B + ∠C = 360^{0}.

**Properties of parallelogram:**

- It has four sides and four vertices and four angles.
- The opposite sides are equal and parallel.
- Each diagonal bisect the parallelogram in two equal triangles.
- The opposite angles are equal.
- The sum of two adjacent angles are equal to 180
^{0}. - Two diagonal bisect each other.
- A parallelogram law: The sum of the square of all the side is equal to the sum of the square of its diagonals.

**What is Trapezium?**

As shown in above figure, A trapezium is a two dimensional plane figure. It is made up of four sides. In which two opposite sides are parallel. Both parallel sides are called the Base and two other sides are called Leg. The distance or length between two parallel line is called the Altitude or Height. It is a convex quadrilateral figure. It is also called Trapezoid.

**Types of Trapezium:**

- Isosceles Trapezium
- Scalene Trapezium
- Right Trapezium

**Isosceles Trapezium:**

A trapezium has non-parallel legs and they are in equal length called Isosceles Trapezium. Like AD = BC. It has two parallel sides. AB||DC.

**Scalene Trapezium:**

A trapezium has all the sides and angles are different in measures is called the Scalene Trapezium. Like Sides: AB ≠ BC; BC ≠ CD; CD ≠ DA; DA ≠ AB; Like Angles: ∠A ≠ ∠D ≠ ∠B ≠ ∠C. It has two parallel sides. AB||DC

**Right Trapezium:**

A trapezium has at least two adjacent right angles and two parallel sides. Here, ∠A = ∠D = 90^{0} and AB||DC.

**The Perimeter of Trapezium:**

The perimeter of trapezium is the sum of the length of all the four sides. It is denoted by capital letter ‘P’.

P = PQ + QR + RS + SP

**Area of Trapezium:**

Area of trapezium is the region covered by its four side in a two dimensional plane figure. It is the product of the average of two parallel sides and its altitudes or height.

Area of trapezium

= ½ (PQ + SR) x (Height)

= ½ (a + b) x h metre^{2}

**Angle of Trapezium:**

The sum of all the four angles are equal to 360^{0}. ∠A + ∠D + ∠B + ∠C = 360^{0}.

**Properties of Trapezium:**

- At least two opposite sides are parallel.
- ∠A + ∠B + ∠C + ∠D = 360°
- Non-parallel sides are not equal. But in case of isosceles trapezium, Non-parallel sides are equal.
- The sum of the two adjacent angles is equal to 180°.
- The mid-point of non-parallel sides are joined together are called the mid-segment. Which is parallel to the other two parallel sides. The mid-segment is equal to the average of two parallel sides.
- The altitude are the distance between two parallel sides.
- Area = ½ (The sum of two parallel sides) x (its altitude)

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