Answer:
See Below.
Step-by-step explanation:
We are given that A, B, C, and D are the midpoints of sides PQ, QR, RS, and SP, respectively.
And we want to prove that ABCD is a parallelogram.
By the definition of midpoint, this means that:
[tex]SD\cong DP, \, PA\cong AQ, \, QB\congBR, \, \text{ and } RC\cong CS[/tex]
To prove, we can construct a segment from S to Q to form SQ. This is shown in the first diagram.
By the Midpoint Theorem:
[tex]DA\parallel SQ[/tex]
Similarly:
[tex]CB\parallel SQ[/tex]
By the transitive property for parallel lines:
[tex]DA\parallel CB[/tex]
Likewise, we can do the same for the other pair of sides. We will construct a segment from P to R to form PR. This is shown in the second diagram.
By the Midpoint Theorem:
[tex]AB\parallel PR[/tex]
Similarly:
[tex]DC\parallel PR[/tex]
So:
[tex]AB\parallel DC[/tex]
This yields:
[tex]DA\parallel CB\text{ and } DC\parallel AB[/tex]
By the definition of a parallelogram, it follows that:
[tex]ABCD\text{ is a parallelogram.}[/tex]
