Question

Suppose two adjacent angles are supplementary. Show that if one of them is an obtuse angle, then the other angle must be acute.

Answer 1

Let $x$ and $y$ be two adjacent supplementary angles
$\Rightarrow x+y={180}^{\circ }$
$\Rightarrow x=180-y$ .......(i)
$\angle x$ is obtuse.
So, ${90}^{\circ }<x<{180}^{\circ }$
Substitute equation(i) in above inequality, we get
$\Rightarrow {90}^{\circ }<{180}^{\circ }-y<{180}^{\circ }$
Subtracting ${180}^{\circ }$, we get
$\Rightarrow {90}^{\circ }-{180}^{\circ }<{180}^{\circ }-{180}^{\circ }-y<{180}^{\circ }-{180}^{\circ }$
$\Rightarrow -{90}^{\circ }<-y<0^{\circ }$
$\Rightarrow 0^{\circ }<y<{90}^{\circ }$ [ multiplying $-1$ ]

Hence, the other angle $y$ is an acute angle.