Given: ΔMNP, ΔMNQ are isosceles
To prove:  ΔMQP ≅ ΔNQP

 


 PM ≅ PN                   Legs of isosceles ΔMNP are ≅

∠PMN ≅ ∠PNM               Base angles of isosceles ΔMNP are ≅

∠QMN ≅ ∠QNM               Base angles of isosceles ΔMNQ are ≅ 

∠PMN+∠QMN ≅ ∠PNM+∠QNM    ≅∠s added to ≅∠s are ≅

 ∠PMQ ≅ ∠PNQ              A whole is equal to the sum of its parts

 QM ≅ QN                   Legs of isosceles ΔMNQ are ≅  

ΔMQP ≅ ΔNQP                SAS

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However there is another case to prove, when Q is on the
same side of MN as P.  This case:

 

In this case, the main difference is that you'll need to 
subtract angles instead of adding them in the 4th step. 


Edwin