Mathematics

State true or false:
The bisectors of two adjacent supplementary angles include a right angle.


ANSWER

True


SOLUTION
$$
We\quad know\quad -\\ Two\quad angles\quad whose\quad sum\quad is\quad { 180 }^{ o }\quad are\quad said\quad to\quad be\quad supplementary.\\ A\quad line\quad which\quad divides\quad any\quad angle\quad into\quad equal\quad parts\quad is\quad called\quad as\quad the\quad bisector\quad of\quad that\quad angle.\\ From\quad the\quad figure,\quad \angle ACD\quad \& \quad \angle DCB\quad are\quad two\quad adjacent\quad angles\quad which\quad are\quad supplementary.\\ \angle ACD+\angle DCB={ 180 }^{ o }\quad \qquad ----\left( 1 \right) \\ Let\quad EC\quad \& \quad CF\quad be\quad the\quad angle\quad bisectors\quad of\quad \angle ACD\quad \& \quad \angle DCB\quad respectively.\\ So\quad \angle AEC=\angle ECD\quad \& \quad \angle AEC+\angle ECD=\angle ACD\Rightarrow 2\angle ECD=\angle ACD\quad \quad ----\left( 2 \right) \qquad \\ Also\quad \angle DCF=\angle FCB\quad \& \quad \angle DCF+\angle FCB=\angle DCB\Rightarrow 2\angle DCF=\angle DCB\quad ----\left( 3 \right) \\ Using\quad \left( 2 \right) \quad \& \quad \left( 3 \right) \quad in\quad \left( 1 \right) \\ 2\angle ECD+2\angle DCF={ 180 }^{ o }\Rightarrow 2\left( \angle ECD+\angle DCF \right) ={ 180 }^{ o }\Rightarrow \angle ECD+\angle DCF=\frac { { 180 }^{ o } }{ 2 } \Rightarrow \angle ECD+\angle DCF={ 90 }^{ o }\Rightarrow \angle ECF={ 90 }^{ o }\quad \left[ From\quad figure \right] \\ Here\quad \angle ECF\quad is\quad the\quad angle\quad included\quad by\quad the\quad angle\quad bisectors.\\ Hence\quad the\quad bisectors\quad of\quad two\quad adjacent\quad supplementary\quad angles\quad always\quad include\quad a\quad right\quad angle.
$$
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TRUE/FALSE Medium Published on 09th 09, 2020
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