Question

In the alongside diagram, $ABCD$ is a parallelogram, then $AB=2BC$

• A. True
• B. False

Answer 1

The correct option is A True

$GivenABCDisaparallelogram,AD\parallel BC\&AB\parallel DC,\angle CBE=\angle EBA=\angle 1,\angle DAE=\angle EAB=\angle 2Construction-DrawEFfromEsuchthatEF=BC\&EF\parallel BC.SoBCEFisaparallelogram.SinceEF=BC\&EF\parallel BCsoEF=AD\&EF\parallel AD.Now\angle CBE=\angle BEF=\angle 1\{AlternateangleswhenEF\parallel BC,BEisthetransversal\}So\angle BEF=\angle EBF=\angle 1\Rightarrow △EFBisaisoscelestriangle\left\{Sincebaseanglesareequal\right\}\therefore in△EFB,EF=FB\Rightarrow FB=BC---\left(1\right)Now\angle FEA=\angle EAD=\angle 2\{AlternateangleswhenEF\parallel AD,EAisthetransversal\}So\angle FEA=\angle EAF=\angle 2\Rightarrow △EABisaisoscelestriangle\left\{Sincebaseanglesareequal\right\}\therefore in△EAB,EF=AF\Rightarrow AF=BC---\left(2\right)Adding\left(1\right)\&\left(2\right),wegetFB+AF=BC+BC\Rightarrow AB=2BC$