Question

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

prove: △pmk ≅ △nqm

1. ∠p ≅ ∠n
2. m of pn
3. m is the midpoint

4.
5.
1.
2.
3.
4.
5.

Explanation:

Step1: Definition of midpoint

Since \(M\) is the mid - point of \(PN\), by the definition of a 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) congruence criterion

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

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 (Angle - Side - Angle) congruence criterion