# Quadrilaterals

## Exercise 8.1

Question 1: The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

**Answer:** As you know angle sum of a quadrilateral = 360°

So, `3x + 5x + 9x + 13x = 360°`

Or, `30x = 360°`

Or, `x = 12°`

Hence, `3x = 36°`, `5x = 60°`, `9x = 108°` and `13x = 156°`

Question 2: If the diagonals of a parallelogram are equal, then show that it is a rectangle.

**Answer;** In the following parallelogram both diagonals are equal:

So, ΔABC ≅ ΔADC ≅ ΔABD ≅ ΔBCD

Hence, ∠A=∠B=∠C=∠D=90°

As all are right angles so the parallelogram is a rectangle.

Question 3: Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

**Answer;** In the given quadrilateral ABCD diagonals AC and BD bisect each other at right angle. We have to prove that `AB=BC=CD=AD`

In ΔAOB and ΔAOD

`DO=OB` (O is the midpoint)

`AO=AO` (common side)

`∠AOB=∠AOD` (right angle)

So, `ΔAOB≅ ΔAOD`

So, `AB=AD`

Similarly `AB=BC=CD=AD` can be proved which means that ABCD is a rhombus.

Question 4: Show that the diagonals of a square are equal and bisect each other at right angles.

**Answer:** In the figure given above let us assume that

`∠DAB=90°`

So, `∠DAO=∠BAO=45°`

Hence, `∠AOD=90°`

`DO=AO` (Sides opposite equal angles are equal)

Similarly `AO=OB=OC` can be proved

This gives the proof of diagonals of square being equal.

#### Key Points About Quadrilaterals

- Sum of the angles of a quadrilateral is 360°.
- A diagonal of a parallelogram divides it into two congruent triangles.
- In a parallelogram,
- opposite sides are equal
- opposite angles are equal
- diagonals bisect each other

- A quadrilateral is a parallelogram, if
- opposite sides are equal or
- opposite angles are equal or
- diagonals bisect each other or
- a pair of opposite sides is equal and parallel

- Diagonals of a rectangle bisect each other and are equal and vice-versa.
- Diagonals of a rhombus bisect each other at right angles and vice-versa.
- Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
- The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.
- A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
- The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.