Suppose thatABCDis a Saccheri quadrilateral and that diagonalsACandBDintersect at pointP. IfMis the midpoint ofDPandNis the midpoint ofCP, prove thatAMis congruent toBN.

Proof. Suppose thatABCDis a Saccheri quadrilateral and that diagonalsACandBDintersect at pointP. AssumeMis the midpoint ofDPandNis the midpoint ofCP. We shall prove thatAMis congruent toBN. By SAS, ∆ADBis congruent to ∆BCA. Therefore, ÐPABis congruent to ÐPBAby CPCTC. Also, line segmentPAis congruent to line segmentPBby the Converse of the Isosceles Triangle Theorem. We can now say that ∆DAPis congruent to ∆CBPby SAS. Therefore, ÐDAPis congruent to ÐCBPby CPCTC and line segmentCPis congruent to line segmentDPby CPCTC.This allows us to say that ∆CPandDPare congruent. We are given thatDMis congruent toMPandCNis congruent toNPby the definition of midpoint. Therefore, we can say thatMPis congruent toNPby the transitive property.APMis congruent to ∆BPNby SAS. Therefore,AMis congruent toBNby CPCTC.

The part of my proof that worries me is thered underlined. Is it ok?