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.

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.