In the given figure, ABCD is a quadrilateral in which AD`=` BC and `/_ ADC = /_ BCD`. Show that the points A,B,C,D lie on a circle.

Explanation

Draw DE `_|_` AB and CF `_|_ ` AB.
In `Delta ADE` and `Delta BCF`, we have
`/_ADE = /_ ADC - 90^(@) implies /_ BCD - 90^(@) = /_ BCF [ :' /_ ADC = /_ BCD ` ( given) ],
and `AD=BC` [ given ]
`:. Delta ADE ~= Delta BCF` [ by RHS -congurence]
`implies /_ A = /_ B`
`:. /_ A + /_ B + /_ C + /_ D = 360^(@)`
`implies 2/_ B + 2 /_ D = 360^(@) [ :' /_ C = /_ D` ( given ) ]
`implies /_ B + /_ D = 180^(@)` and therefore, `/_ A + /_ C = 180^(@)`
Hence, the points A,B,C,D are concyclic.