We know that triangle MNQ is isosceles with base line segment MQ So, line segment MN is congruent to line segment QN by the definition of isosceles triangle. The base angles of the isosceles triangle, angle NMS and angle NQS , are congruent by the isosceles triangle theorem. It is also given that line segment NR and line segment MQ bisect each other at cap s Segments _______ are therefore congruent by the definition of bisector. Thus, triangle MNS is congruent to triangle QNS by SAS.
Since segments NR and MQ bisect each other at S, S is the midpoint of MQ. Therefore, MS and QS are congruent by the definition of a bisector. Together with MN ≅ QN and the congruent base angles, this gives SAS for triangles MNS and QNS.
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