Question

1. given: ∠p ≅ ∠n and m is the midpoint of pn.

prove: △pmk ≅ △nqm

1. ∠p ≅ ∠n
2. of pn.
3. ∠pmk ≅ ∠nmq

4.

1. △pmk ≅ △nqm

1.
2.

1. vertical angles theorem

4.
5.

Explanation:

Step1: Definition of midpoint

Since \(M\) is the mid - point of \(PN\), by the definition of mid - point, \(PM = NM\).

Step2: Vertical angles

\(\angle PMK\) and \(\angle NMQ\) are vertical angles. By the vertical angles theorem, \(\angle PMK=\angle NMQ\).

Step3: ASA (Angle - Side - Angle) criterion

We are given \(\angle P=\angle N\), \(PM = NM\) (from Step 1), and \(\angle PMK=\angle NMQ\) (from Step 2). By the ASA (Angle - Side - Angle) congruence criterion for triangles, \(\triangle PMK\cong\triangle NQM\).

Answer:

1. \(\angle P=\angle N\) (Given)
2. \(PM = NM\) (Definition of mid - point)
3. \(\angle PMK=\angle NMQ\) (Vertical angles theorem)
4. \(\triangle PMK\cong\triangle NQM\) (ASA)
5. \(\triangle PMK\cong\triangle NQM\) (Conclusion)