ABCD is a quadrilateral in which AB=BC and AD =CD ,Show that BD bisects boht the angle ABC and ADC

Explanation

Given ABCD is a quadrilateral in which AB=BC and AD = CD .
TO show BD bisects boht the anlges ABC and ADC
Proof Since AD = BC (given )

`therefore " " angle2=angle1 " "(i )`
[angle opposite to equal sides are equal ]
and AD =CD [given ]
`rArr " " angle 4 = angle 3 " "..(i)`
On adding Eqs.(i) and( ii) we get
`angle 2 + angle 4=angle 1 + angle 3 `
`rArr " " angle BCD = angle BAD ...(iii)`
in `Delta BAD and Delta BCD , `
AB=BC [given ]
`angle BAD = angle BCD " "` [from Eq.(ii) ]
AD =CD [given ]
`therefore " " Delta BAD cong Delta BCD " " ` [by SAS congruence rule ]
Hence,`angle ABD = angle CBD " and " angle ADB= angle CDB ` i.e., BD bisects the angle ABC and ADC
[ by CPCT ]