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masters Hello,
Given:
$\displaystyle \overline{NQ}$ bisects $\displaystyle \angle MNP$ and $\displaystyle \angle M \cong \angle P$
Reflexive Property of Congruence:
$\displaystyle \overline{NQ} \cong \overline {NQ}$
Definition of Angle Bisector
$\displaystyle \angle MNQ \cong \angle PNQ$
AAS: (Angle-Angle-Side Congruency Theorem)
$\displaystyle \triangle MNQ \cong \triangle PNQ$
CPCTC (Corresponding Parts of Congruent Trinagles are Congruent)
$\displaystyle \overline{MQ} \cong \overline{PQ}$