Question

State true or false:

In the alongside diagram, $ABCD$ is a parallelogram in which $AP$ bisects angle $A$ and $BQ$ bisects angle $B$, then

$AQ=BP$

• A. True
• B. False

Answer 1

The correct option is A True

$GivenABCDisaparallelogram;\angle PBQ=\angle QBA=\angle 1\&\angle QAP=\angle PAB=\angle 2Fromthefigure,\angle PBQ=\angle AQB=\angle 1\{AQ\parallel BP\&BQisthetransversal\}So\angle AQB=\angle QBA=\angle 1\Rightarrow △BAQisanisoscelestriangle\{sincebaseanglesareequal.\}\therefore in△BAQ,SideAQ=SideAB---\left(1\right)Againfromthefigure,\angle QAP=\angle APB=\angle 2\{AQ\parallel BP\&APisthetransversal\}So\angle PAB=\angle APB=\angle 2\Rightarrow △ABPisanisoscelestriangle\{sincebaseanglesareequal.\}\therefore in△ABP,SideBP=SideAB---\left(2\right)From\left(1\right)\&\left(2\right),wegetAQ=BP$