Final answer:
Triangle MQS is congruent to triangle PNR in quadrilateral ABCD because we can establish the congruence of two sides and the included angle, allowing the use of the Angle-Side-Angle congruence theorem.
Step-by-step explanation:
To prove that triangle MQS is congruent to triangle PNR within quadrilateral ABCD, where M, N, P, Q, R, S are the midpoints of AB, BC, CD, DA, AC, and BD, we follow these steps:
- Segment MQ is parallel to segment PR (because they are both midsegments of the triangles formed by the diagonals of the quadrilateral), and MQ = 1/2 of PR by the properties of midpoints.
- Segment QS is parallel to segment NR (similarly, they are midsegments) and QS = 1/2 of NR again by the properties of midpoints.
- Angle MQS is equal to angle PNR because they are both vertical angles and as such are always equal.
By establishing the congruence of two sides and the included angle, we can use the Angle-Side-Angle congruence theorem to determine that triangle MQS is congruent to triangle PNR.
The properties of midpoints for sides, the Angle-Side-Angle congruence theorem for angles and sides, and the equality of vertical angles together supply the proof without requiring the Pythagorean theorem or the Law of Sines. These latter two are not applicable in this context as we are not dealing with a right triangle or a need to relate the sides and angles in non-right triangles.