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

Prove that RSTU must be a parallelogram.

In the diagram shown parallelogram MNPQ is shown with points located on its such that MU SP and RN QT Prove that RSTU must be a parallelogram class=

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Answer:

See explanation

Step-by-step explanation:

MNPQ is a parallelogram. Then

  • [tex]MQ\cong NP;[/tex]
  • [tex]MN\cong QP.[/tex]

By segment addition postulate,

  • [tex]MQ\cong QU+UM;[/tex]
  • [tex]PN\cong PS+SN;[/tex]
  • [tex]QP\cong QT+TP;[/tex]
  • [tex]MN\cong MR+RN.[/tex]

Given that MU ≅ SP and RN ≅ QT, then [tex]QU\cong SN[/tex] and [tex]MR\cong TP.[/tex]

Consider triangles QUT and NSR. In these triangles,

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

By SAS postulate, triangles QUT and NSR are congruent. Then [tex]UT\cong SR.[/tex]

Consider triangles MRU and PTS. In these triangles,

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

By SAS postulate, triangles MRU and TPS are congruent. Then [tex]UR\cong TS.[/tex]

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

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