Question

given the figure, where ∠pmn is congruent to ∠qmn, and ∠p is congruent to ∠q, what can be concluded? ∠pmn ≅ ∠mpn, m∠p + m∠q = 180°, mq = qn, △mpn ≅ △mqn

Explanation:

Step1: Identify Common Side

In triangles \( \triangle MPN \) and \( \triangle MQN \), \( MN \) is a common side, so \( MN = MN \) (reflexive property).

Step2: List Given Congruences

We know \( \angle PMN \cong \angle QMN \) and \( \angle P \cong \angle Q \).

Step3: Apply AAS Congruence

By the Angle - Angle - Side (AAS) congruence criterion (two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle), in \( \triangle MPN \) and \( \triangle MQN \):

• \( \angle P\cong\angle Q \)
• \( \angle PMN\cong\angle QMN \)
• \( MN = MN \)

So, \( \triangle MPN\cong\triangle MQN \) by AAS.

Let's analyze the other options:

• For \( \angle PMN\cong\angle MPN \): There is no information given to support this congruence of angles.
• For \( m\angle P + m\angle Q=180^{\circ} \): We know \( \angle P\cong\angle Q \), but there is no indication that they are supplementary.
• For \( MQ = QN \): There is no information to suggest that \( MQ \) and \( QN \) are congruent.

Answer:

\( \triangle MPN \cong \triangle MQN \)