## Pre-Requisires

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**Quadrilateral**

Any closed polygon with four sides, four angles and four vertices are called **Quadrilateral**. It could be regular or irregular. **(Sroll down to continute …)**

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**Quadrilateral**

Quadrilateral is a closed figure with four sides.

**Characteristics of a quadrilateral**

**Angle Sum Property of a Quadrilateral: **

Qudrilateral is a four sided closed figure.

Sum of all angles of a quadrilateral is 360°.

**Types Of Quadrilaterals**

Quadrilaterals are broadly classified into three categories as:

(i) Kite

(ii) Trapezium

(ii) Parallelogram

**Kite:**

(i) Kite has no parallel sides

(ii) Kite has a pair of equal adjacent sides.

(ii) It is not a parallelogram

**Characteristics Of Kite:**

**Perimeter ****Of Square**

**Of Square**

**Area Of Kite**

**Trapezium:**

Trapezium is a quadrilateral with the following characteristics:

(i) One pair of opposite sides is parallel to each other.

(ii) The other pair of opposite sides may not be parallel to each other.

**Characteristics Of Trapezium**

(i) Sum of all angles of a quadrilateral is 360°.

(ii) One pair of opposite sides is parallel to each other.

(iii) The other pair of opposite sides need not be parallel to each other.

**Types Of Trapezium:**

Quadrilaterals are broadly classified into two categories as:

(i) Isosceles Trapezium.

(ii) Scalene Trapezium.

(i) Right Trapezium.

**Isosceles Trapezium:**

Isosceles Trapezium is a quadrilateral with the following characteristics:

(i) One pair of opposite sides is parallel to each other.

(ii) The other pair of opposite sides are equal.

(iii) The other pair of opposite sides need not be parallel to each other.

Isosceles Trapezium is a trapezium with the following characteristics:

(i) One pair of opposite sides is parallel to each other.

(ii) The other pair of opposite sides are equal.

(iii) The other pair of opposite sides need not be parallel to each other.

**Characteristics Of Isosceles Trapezium**

(i) Sum of all angles of a quadrilateral is 360°.

(ii) One pair of opposite sides is parallel to each other.

(iii) The other pair of opposite sides are equal.

(iv) The other pair of opposite sides need not be parallel to each other.

**Scalene Trapezium:**

- Scalene trapezium: Classified by the length of the legs or the measurement of their angles.

**Characteristics Of Scalene Trapezium**

**Right Trapezium:**

- Right trapezium: Has one pair of parallel sides and one pair of right angles.

**Characteristics Of Right Trapezium**

**Perimeter Of ****Trapezium**

**Trapezium**

**Area Of ****Trapezium**

**Trapezium**

**Parallelogram:**

Parallelogram is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

**Characteristics of a parallelogram**

(i) Sum of all angles of a Parallelogram is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

(ii) Two pairs of opposite angles are equal.

(iii) Diagonals bisect each other.

(iv) Diagonals need not be equal to each other.

(v) Diagonals divide it into two congruent triangles.

**Types Of Parallelogram**

Parallelograms are broadly classified into three categories as:

(i) Rectangle

(ii) Rhombus

(iii) Square

**Perimeter Of ****Parallelogram**

**Parallelogram****Area Of ****Parallelogram**

**Parallelogram****Rectangle:**

Rectangle** **is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

(iii) All four angles are right angles. (each angle is 90 ^{o}).

**Characteristics Of Rectangle **

(i) Sum of all angles of a quadrilateral is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(ii) Two pairs of opposite sides are equal in length.

(iii) All four angles are right angles. (each angle is 90 ^{o}).

(iii) Diagonals bisect each other.

(iv) Diagonals are equal to each other.

(v) Diagonals of a rectangle divide it into two congruent triangles.

**Conclusions**:

- Every Rectangle is a Parallelogram. But Every Parallelogram need not to be a Rectangle.

**Condition for a rhombus to be a square:**

If all four angles of a parallelogram are right angles. (each angle is 90 ^{o}), the parallelogram becomes a Rectangle.

**Perimeter Of Rectangle**

**Area Of Recatangle **

**Rhombus:**

Rhombus is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) All four sides are equal in length.

**Characteristics Of Rhombus**

(i) Sum of all angles of a quadrilateral is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(ii) All four sides are equal in length.

(ii) Two pairs of opposite angles are equal.

(iii) Diagonals bisect each other.

(iv) Diagonals need not be equal to each other.

(v) Diagonals divide a Rhombus into two congruent triangles.

**Conclusions**:

- Every Rhombus is a Parallelogram. But Every Parallelogram need not to be a Rhombus.

**Condition for a rhombus to be a square:**

If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.

**Perimeter Of Rhombus**

**Area Of ****Rhombus**

**Rhombus**

**Square:**

Square is a quadrilateral with the following characteristics:

(i) Two pairs of opposite sides are parallel to each other.

(ii) All four sides are equal in length.

(iii) All four angles are right angles. (each angle is 90 ^{o}).

**Characteristics Of Square**

(i) Sum of all angles of a quadrilateral is 360°.

(ii) Two pairs of opposite sides are parallel to each other.

(iii) All four sides are equal in length.

(iv) All four angles are right angles. (each angle is 90 ^{o}).

(v) Diagonals bisect each other.

(vi) Diagonals need not be equal to each other.

(vii) Diagonals divide a Rhombus into two congruent triangles.

**Conclusions**:

- Every square is a Rhombus. But Every Rhombus need not to be a square.

**Condition for a rhombus to be a square:**

If all the angles of a rhombus are right angles (euqal to 90^{o}), the rhombus becomes a square.

2. Every Square is a prallelogram. But Every prallelogram need not to be a square.

**Condition for a prallelogram** **to be a square:**

(i) If all the angles of a parallelogram are right angles (euqal to 90^{o}), and all the sides of a parallelogram are equal in length, the parallelogram becomes a square.

3. Every Square is a rectangle. But Every Rectangle need not to be a square.

**Condition for a Rectangle** **to be a square:**

If all the sides of a Rectangle are equal in length, the Rectangle becomes a square.

If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.

**Perimeter ****Of Square**

**Of Square**

**Area Of Square**

**Important Points To Remember**

- The diagonals of a parallelogram are equal if and only if it is a rectangle.
- If a diagonal of a parallelogram bisects one of the angles of the parallelogram then it also bisects the opposite angle.
- In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.
- The angle bisectors of a parallelogram form a rectangle.

**Mid Point Theorem**

A line segment joining the mid points of any two sides of a triangle is parallel to the third side and length of the line segment is half of the parallel side.

**Converse Of Mid Point Theorem**

A line through the midpoint of a side of a triangle parallel to another side bisects the third side.

**Intercept Theorem**

If there are three parallel lines and the intercepts made by them on one transversal are equal then the intercepts on any other transversal are also equal.

**Angle Sum Property of a Quadrilateral**

The sum of the four angles of a quadrilateral is 360°

If we draw a diagonal in the quadrilateral, it divides it into two triangles.

And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.

The sum of angles of ∆ADC = 180°.

The sum of angles of ∆ABC = 180°.

By adding both we get ∠A + ∠B + ∠C + ∠D = 360°

Hence, the sum of the four angles of a quadrilateral is 360°.

**Example**

Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.

**Solution:**

Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.

So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).

∠B = 52°

∠A = 180°- 52° = 128°

∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).

∠C = 60°

128° + 52° + 60° + ∠D = 360°

∠D = 120°

∴ ∠A = 128° and ∠D = 120 °.

**Types of Quadrilaterals**

S No. | Quadrilateral | Property | Image |

1. | Kite | a. No Parallel Sides b. Two pairs of adjacent sides are equal. | |

2. | Trapezium | One pair of opposite sides is parallel. | |

3. | Parallelogram | Both pairs of opposite sides are parallel. | |

3. | Rectangle | a. Both the pair of opposite sides are parallel. b. Opposite sides are equal.c. All the four angles are 90°. | |

4. | Square | a. All four sides are equal. b. Opposite sides are parallel. c. All the four angles are 90°. | |

5. | Rhombus | a. All four sides are equal. b. Opposite sides are parallel. c. Opposite angles are equal.d. Diagonals intersect each other at the centre and at 90°. |

**Remark**: A square, Rectangle and Rhombus are also a parallelogram.

**Properties of a Parallelogram**

**Theorem 1**: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.

∆ABD ≅ ∆CDB

**Theorem 2**: In a parallelogram, opposite sides will always be equal.

**Theorem 3**: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.

Here, AD = BC and AB = DC

Then ABCD is a parallelogram.

**Theorem 4**: In a parallelogram, opposite angles are equal.

In ABCD, ∠A = ∠C and ∠B = ∠D

**Theorem 5**: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.

**Theorem 6**: The diagonals of a parallelogram bisect each other.

Here, AC and BD are the diagonals of the parallelogram ABCD.

So the bisect each other at the centre.

DE = EB and AE = EC

**Theorem 7**: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.

This is the reverse of the theorem 6.

**The Mid-point Theorem**

1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.

If AB = BC and CD = DE then BD ∥ AE.

2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line.

If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.

**Example**

Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.

**Solution:**

Let BE cut DC at a point G.

Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.

By midpoint theorem, G is the midpoint of EB.

Again in ∆BEF, G is the midpoint of BE and GC∥ EF.

So, by midpoint theorem C is the midpoint of BF.

Hence proved.

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