Question

In the diagram shown, parallelogram MNPQ is shown with points located on it's such that MU ≅ SP and RN ≅ QT.

Answer 1

Answer:

See explanation

Step-by-step explanation:

MNPQ is a parallelogram. Then

• $MQ\cong NP;$
• $MN\cong QP.$

By segment addition postulate,

• $MQ\cong QU+UM;$
• $PN\cong PS+SN;$
• $QP\cong QT+TP;$
• $MN\cong MR+RN.$

Given that MU ≅ SP and RN ≅ QT, then $QU\cong SN$ and $MR\cong TP.$

Consider triangles QUT and NSR. In these triangles,

• $QU\cong NS$ - proven;
• $QT\cong NR$ - given;
• $\angle UQT\cong \angle SNR$ - as opposite ngles of parallelogram MNPQ.

By SAS postulate, triangles QUT and NSR are congruent. Then $UT\cong SR.$

Consider triangles MRU and PTS. In these triangles,

• $MR\cong PT$ - proven;
• $MU\cong PS$ - given;
• $\angle UMR\cong \angle SPT$ - as opposite ngles of parallelogram MNPQ.

By SAS postulate, triangles MRU and TPS are congruent. Then $UR\cong TS.$

In quadrilateral RSTU, $UT\cong SR$ and $UR\cong TS.$ Since opposite sides of the quadrilateral are congruent, the quadrilateral is a parallelogram.