Given: M is the midpoint of line segment PQ and line segment RS.

Prove: line segment QR is congruent to line segment PS.

To prove that line segment QR is congruent to line segment PS, we will use the midpoint theorem.

The midpoint theorem states that if M is the midpoint of line segment AB, then line segment AB is divided into two congruent segments.

Since M is the midpoint of line segment PQ and line segment RS, we can say that line segment PQ is divided into two congruent segments MQ and MP, and line segment RS is divided into two congruent segments MS and MR.

Using the transitive property of equality, we can conclude that MQ is congruent to MS and MP is congruent to MR.

Next, we can use the segment addition postulate, which states that if three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.

Applying the segment addition postulate, we have:
MQ + QR = MR
MP + PS = MS

Since MQ is congruent to MS and MP is congruent to MR, we can substitute these values into the equations above:
MS + QR = MR
MR + PS = MS

Using the segment addition postulate again, we can combine the equations:
QR + PS = MR

By the properties of equality, we can switch the order of addition:
PS + QR = MR

Finally, by the symmetric property of equality, we can conclude that QR is congruent to PS:
QR ≅ PS

Therefore, we have proven that line segment QR is congruent to line segment PS, using the midpoint theorem and segment addition postulate.

To prove that line segment QR is congruent to line segment PS when M is the midpoint of line segment PQ and line segment RS, we can use the midpoint theorem.

The midpoint theorem states that if a line segment has a midpoint, then the line segment is divided into two congruent segments.

Here's how we can prove it:

1. Given that M is the midpoint of line segment PQ and line segment RS.

2. By the midpoint theorem, we know that PM is congruent to MQ, and RM is congruent to MS.

3. Since PM is congruent to MQ, we can substitute MQ with PM in the line segment QR. This gives us QR is congruent to PR.

4. Similarly, since RM is congruent to MS, we can substitute MS with RM in the line segment PS. This gives us PS is congruent to PR.

5. Therefore, we have QR is congruent to PR and PS is congruent to PR.

6. By the transitive property of congruence, if QR is congruent to PR, and PS is congruent to PR, then QR is congruent to PS.

Thus, we have proved that line segment QR is congruent to line segment PS.

It's important to note that this proof relies on the use of the midpoint theorem.

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