Question

State true or false:

The bisectors of two adjacent supplementary angles include a right angle.

• A. True
• B. False

Answer 1

The correct option is A True

$Weknow-Twoangleswhosesumis{180}^{o}aresaidtobesupplementary.Alinewhichdividesanyangleintoequalpartsiscalledasthebisectorofthatangle.Fromthefigure,\angle ACD\&\angle DCBaretwoadjacentangleswhicharesupplementary.\angle ACD+\angle DCB={180}^o----\left(1\right)LetEC\&CFbetheanglebisectorsof\angle ACD\&\angle DCBrespectively.So\angle AEC=\angle ECD\&\angle AEC+\angle ECD=\angle ACD\Rightarrow 2\angle ECD=\angle ACD----\left(2\right)Also\angle DCF=\angle FCB\&\angle DCF+\angle FCB=\angle DCB\Rightarrow 2\angle DCF=\angle DCB----\left(3\right)Using\left(2\right)\&\left(3\right)in\left(1\right)2\angle ECD+2\angle DCF={180}^o\Rightarrow 2(\angle ECD+\angle DCF)={180}^o\Rightarrow \angle ECD+\angle DCF=\frac{{180}^o}{2}\Rightarrow \angle ECD+\angle DCF={90}^o\Rightarrow \angle ECF={90}^o\left[Fromfigure\right]Here\angle ECFistheangleincludedbytheanglebisectors.Hencethebisectorsoftwoadjacentsupplementaryanglesalwaysincludearightangle.$