Chapter 8: Quadrilaterals

EXERCISE 8.1

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

Solution: let 3x , 5x ,9x and  13x are the angle of quadrilateral respectively .

We know that , the sum of the angles of a quadrilateral is 360° .

 3x+5x+9x+13x=360°

 ⇒30x=360°

⇒x=360°30

⇒x=12°

3x=3×12°=36°  

 5x=5×12°=60°

9x=9×12°=108°  

And 13x=13×12°=156°

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

Solution: let ABCD is a parallelogram and AC = BD . Then we show that ABCD is a rectangle .

Proof : Since, ABCD is a parallelogram , then

AB = CD  and AD = BC

In ∆ADB and  ∆ABC , we have

AB=AB  [Given]

  AD=BC  [Given]

  BD=AC  [Given]

∆ADB ≅∆ABC [SSS]

∠BAD=∠ABC   [CPCT]

Again, AD∥BC and  AB is a transversal .

∠BAD+∠ABC=180°

 ∠ABC+∠ABC=180°

2∠ABC=180°

∠ABC=180°2

∠ABC=90°

So, ABCD is a parallelogram in which one angle is 90° .

Therefore, ABCD is a rectangle .

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

Solution:  let ABCD is a quadrilateral such that  OA=OC , OB=OD and ∠AOB=∠BOC=∠COD=∠AOD=90° . AC and BD are the diagonals . Then we show that quadrilateral ABCD is rhombus .

Proof: In∆AOB and  ∆AOD ,we have

 OB=OD

  ∠AOB=∠AOD

  OA=OA

  ∆AOB≅∆AOD  [SAS]

AB=AD [CPCT]

Similarly, AB=BC ,  BC=CD and AD=CD

So,  AB=BC=CD=AD

Therefore, the quadrilateral ABCD is rhombus .