4. Special types of parallelogram

Special types of parallelogram

• Concept of trapezium

A quadrilateral with one pair of parallel side is called trapezium.

A trapezium is isosceles trapezium, if its non-parallel sides are equal.

In a trapezium, the following properties are true:

1. The sum of all the four angles of the trapezium is equal to 360°.
2. A trapezium has 4 unequal sides.
3. A trapezium has two parallel sides and two non-parallel sides.
4. The diagonals of trapezium bisect each other.
5. The length of the mid-segment is equal to 12 the sum of the parallel bases, in a trapezium.
6. Sum of adjacent angles on non-parallel sides of trapezium is 180°.

a quadrilateral ABCD is an isosceles trapezium, if AD||BC and AB=CD.

A quadrilateral is a parallelogram if its both pairs of opposite sides are parallel.

• Concept of kite

A quadrilateral with two pairs of equal adjacent sides and unequal opposite sides is called a kite.

define a kite ABCD is a quadrilateral having two pairs of equal adjacent sides and unequal opposite sides.

Thus, if ABCD is a kite, then AB=AD and BC=CD.

In a kite, the following properties are true:

1. The sum of all the four angles of the kite is equal to 360°.
2. A kite has two pairs of equal adjacent sides.
3. A kite has unequal opposite sides.
4. The diagonals of kite perpendicular bisect of each other.
5. One of the diagonals bisects the other diagonal.
6. In the figure, ∠B=∠D but \(∠A ≠ ∠C\).

• Concept of rhombus

A parallelogram with all sides are equal is called a rhombus.

Rhombus is a special case of kite. Note that the sides of a rhombus are all of the same length; this is not the case with the kite.

A rhombus has all the properties of a parallelogram and also that of a kite.

In a rhombus, the following properties are true:

1. The sum of all the four angles of the rhombus is equal to 360°.
2. The opposite sides are equal in length.
3. The opposite angles are equal in measure.
4. The adjacent angles are supplementary.
5. The diagonals are perpendicular bisector of each other.

define a rhombus ABCD as a quadrilateral is a parallelogram with all the sides are in equal measures.

Thus, if ABCD is a rhombus then AB=BC=CD=AD, AB||CD and BC||AD.

• Concept of rectangle

A parallelogram whose each angle is a right angle is called a rectangle.

A rectangle has all the properties of a parallelogram with interior.

In a rectangle, the following properties are true:

1. The opposite sides are parallel and equal in length.
2. The interior angles of the rectangle is 90°.
3. The diagonals are equal in measure and bisect each other.
4. The adjacent angles are supplementary.

define a rectangle ABCD as a quadrilateral is a parallelogram whose each interior angle is 90°

Thus, if ABCD is a rectangle then AB=BC=CD=AD and AB||CD and BC||AD.

Important!

• Concept of square

A rectangle is called a square whose adjacent sides are equal.

Rhombus is a special case of kite. Note that the sides of a rhombus are all of the same length; this is not the case with the kite.

A rectangle has all the properties of a parallelogram.

In a square, the following properties are true:

1. The opposite sides are parallel.
2. All the four sides are in equal measure.
3. The interior angles of the square is 90°.
4. The diagonals are equal in measure and perpendicular bisector of each other.
5. The adjacent angles are supplementary.

define a square ABCD as a quadrilateral is a rectangle whose sides are in the same measure.

Thus, if ABCD is a rectangle then AB=BC=CD=AD and AB||CD and BC||AD.