Question

Prove that opposite sides of a quadrilateral circumscribing a circle

Answer 1

GIVEN ;-

⇒ ABCD is a quadrilateral and it has circumscribing a circle Which has centre  O.

CONSTRUCTION ;-

⇒ Join -  AO, BO, CO, DO.

TO PROVE :-

⇒  Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

PROOF ;-

⇒ In the given figure , we can see that
  
                       ⇒  ∠DAO = ∠BAO [Because, AB and AD are tangents in the                                                       circe] 

So , we take this angls as 1 , that is ,
          
                       ⇒  ∠DAO = ∠BAO = 1

Also  in quad. ABCD , we get,
 
                      ⇒ ∠ABO = ∠CBO { Because , BA and BC are tangents }

⇒Also , let us take this angles as 2. that is ,
 
                      ⇒ ∠ABO = ∠CBO = 2 

⇒ As same as , we can take for vertices C and as well as D.

⇒ Sum. of angles of quadrilateral ABCD =  360° { Sum of angles of quad                                                                                   is 360°}

Therfore ,

       ⇒ 2 (1  + 2 + 3 + 4 )  =  360° { Sum. of angles of quad is - 360° }
          
  
        ⇒ 1  +  2  +  3  +  4 = 180°  

Now , in Triangle  AOB,
               
                       ⇒ ∠BOA =  180  –   ( a + b )
                                                                             ⇒ { Equation 1 }
Also , In triangle COD,
 
 
                      ⇒ ∠COD  =  180  –  ( c + d )
                                                                              ⇒ { Equation 2 }

 ⇒From Eq. 1 and 2 we get ,
 
                                ⇒ Angle  BOA + Angle  COD

                                 = 360 – ( a  +  b  +  c  +  d ) 


                                 =  360°   –  180° 


                                 = 180° 

⇒So , we conclude that the line  AB and CD subtend supplementary angles at the centre  O


⇒Hence it is proved that - opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Answer 2

Step-by-step explanation:

mark me as a brainlist please