We are asked to show that the two shaded triangles are congruent in the kite. Let's mark the congruent sides of the kite.
Point N is equidistant from P and M, and point Q is also equidistant from P and M. According to the Converse of the Perpendicular Bisector Theorem, they are both on the perpendicular bisector of PM. This means that R is the midpoint of PM, so segments PR and MR are congruent.
Since NR is a common side of triangles â–ł PNR and â–ł MNR, these two triangles have three pairs of congruent sides. According to the Side-Side-Side (SSS) Congruence Postulate, this means that the two triangles are congruent.
△ PNR≅ △ MNR
We can summarize the steps above in a two-column proof.
Completed Proof
2
&Given:&& PQMN is a kite
& && NP≅NM, QP≅QM
&Prove:&& △ MNR≅△ PNR
Proof:
