If two sides of a cyclic quadrilateral are parallel, prove that:

(i) its other two sides are equal

(ii) its diagonals are equal

Asked by pereiracalida | 14th Jan, 2018, 07:30: PM

Expert Answer:

 
begin mathsize 16px style SInce space ABCD thin space is space straight a space cyclic space quadrilateral comma space opoosite space angles space are space supplementary.
rightwards double arrow angle straight B plus angle straight D equals 180 degree space space space space space space.... left parenthesis straight i right parenthesis
AB space is space parallel space to space CD space and space BC space is space straight a space transversal comma
rightwards double arrow angle ABC plus angle BCD equals 180 degree space space space space space left parenthesis sum space of space interior space angles equals 180 degree right parenthesis
rightwards double arrow angle straight B plus angle straight C equals 180 degree space space space space space space left parenthesis ii right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
angle straight B plus angle straight D equals angle straight B plus angle straight C
rightwards double arrow angle straight D equals angle straight C space space space space space space.... left parenthesis iii right parenthesis
In space triangles space ADC space and space BCD comma space we space have
angle ADC equals angle BCD space space space space space left square bracket From space left parenthesis iii right parenthesis right square bracket
CD equals CD space space space space left parenthesis common right parenthesis
angle DAC equals angle CBD
rightwards double arrow triangle ADC approximately equal to triangle BCD
rightwards double arrow AD equals BC space and space AC equals BD end style

Answered by Rashmi Khot | 15th Jan, 2018, 09:58: AM