Answer variants:
\(\angle OPN\)
bisected angles
two
\(\angle QON\)
common
\(OM\)
\(OM\) is the angle bisector of \(\angle POQ\). \(NP\) and \(NQ\) meet \(OA\) and \(OB\) respectively at \(57^\circ\). Complete the missing fields to prove that the triangles \(OPN\) and \(OQN\) are congruent to each other.
Proof:
We know that
is the angle bisector of \(\angle POQ\).
is the angle bisector of \(\angle POQ\).
Hence, \(\angle PON = \) .
[Since the angles mentioned in the previous step are ]
Now, let us consider the triangles OPN and OQN.
\(= \angle OQN =\) \(57^\circ\) [Given]
Also, \(ON\) is to both the triangles \(OPN\) and \(OQN\)
Here, one corresponding pair of sides and corresponding pair of angles are equal.
Thus by congruence criterion, \(OPN\) \(\cong\) \(OQN\).