Question

in figure OP perpendicular to ray AB and OQ perpendicular to ray AC Prove that AO is bisector of angle CAB​

Answer 1

Congruent Conditions Proved Helpful

Step-by-step explanation:

Let us take Right triangle APO, right-angled at P,

OP is perpendicular to AB or $\angle$APO = 90 °

Now,  

In triangle ΔAQO, right-angled at Q,

QO is perpendicular to AC or $\angle$AQO = 90 °

Taking both triangles ΔAPO and ΔAQO,

AO = AO (Common Side)

$\angle$APO = $\angle$AQO = 90 °

AQ = AP (Perpendicular being drawn from same point O)

Hence, by SAS (Side-Angle-Side Congruence Condition)

ΔAPO ≅ ΔAQO,

So, $\angle$QAO = $\angle$BAO (Corresponding Parts of Congruent triangles )

Or we can conclude that AO is the bisector of \angle QAP,