Question

$ABCD$ is a cyclic quadrilateral with $AB\parallel DC$. Find the relation between the lengths of the non parallel sides $AD$ & $BC$.

• A. $AD=BC$
• B. $AD\ne BC$
• C. $AD>BC$
• D. $AD<BC$

Answer 1

The correct option is A $AD=BC$

$Given-ABCDisacyclicquadrilateralwithAB\parallel DC.Tofindout-therelationbetweenthelengthsofthenonparallelsidesAD\&BC.Solution-WejoinAC\&BD.SinceAB\parallel DC,wehave,\angle ADC+\angle DAB={180}^o........\left(i\right).Again,ABCDisacyclicquadrilateral.\therefore \angle DAB+\angle BCD={180}^o......\left(ii\right).(sincethesumoftheoppositeanglesofacyclicquadrilateral={180}^o).Sofrom\left(i\right)\&\left(ii\right)weget,\angle ADC=\angle BCD.......\left(iii\right).Also,thechordDCsubtends\angle DAC\&\angle DBCtothecircumferenceofthecircleatA\&Brespectively.\therefore \angle DAC=\angle DBC(sinceangles,subtendedbyachordofacircletoitscircumference,areequal)...........\left(iv\right).Nowbetween\Delta ADC\&\Delta BDCwehave,using\left(iii\right)\&\left(iv\right),\angle ADC=\angle BCD,\angle DAC=\angle DBC.Sothethirdangles\angle BDC=\angle ACBandDCiscommonside.\therefore ByASAtest,\Delta ADC\equiv \Delta BDC⟹AD=BC.Hence,optionCiscorrect.$