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**Quadrilaterals | Speed Notes**

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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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**Revision Notes – CBSE 09 Math – Quadrilaterals**

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

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

3. | Rectangle | a. Both the pair of opposite sides is 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°. | |

6. | Kite | Two pairs of adjacent sides are equal. |

**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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