Question

Prove that the bisectors of two adjacent supplementary angles include a right angle.

Answer 1

As they both are supplementary angles,
So, consider the angles as x and y
X+Y=180°
X/2+Y/2=180°/2=90°
So, in /\ AOB
/_A+/_B+/_O=180°
X/2+Y/2+/_O=180°
/_O=90°

Answer 2

Solution:

$\angle SPR+\angle TQP=180^{\circ}$

RP and RQ are bisector of $\angle SPR+\angle TQP$ respectively.

Therefore,

$\angle SPR=\angle RPQ=\frac{1}{2} \angle SPQ \rightarrow (i)$

$\angle TQR=\angle RQP=\frac{1}{2} \angle TQP \rightarrow (ii)$

$\angle SPQ+\angle TQP=180^{\circ} \rightarrow (iii)$

Using (i) and (ii) in (iii),

$2 \angle RPQ+2 \angle RQP=180^{\circ}$

$\angle RPQ+\angle RQP=90^{\circ} \rightarrow (iv)$

In $\Delta PQR$,

$\angle RPQ+\angle RQP+\angle PQR=180^{\circ} \rightarrow(v)$

Apply (iv) in (v),

Then,

$90^{\circ}+\angle PQR=180^{\circ}$

$\angle PQR=90^{\circ}$

Hence Proved.