If point R is the midpoint of both QS and PT, what additional information would you need to state that triangle PQR is congruent to TSR by the SAS Congruence Theorem?

Question

QR is congruent to SR

Angle PRQ is congruent to angle TRS

PQ is congruent to TS

Angle QPR is congrent to angle STR

Answer 1

To state that triangle PQR is congruent to triangle TSR by the SAS (Side-Angle-Side) Congruence Theorem, we need the following pieces of information:

1. One pair of congruent sides: We already have \(PQ \cong TS\) or \(QR \cong SR\) (we'll choose one of these).

2. The angles between those sides: We also need either the angle PRQ or angle TRS to be congruent since it’s the angle between the two sides we are considering for congruency.

3. The second pair of congruent sides: We need to establish that the segment QR is congruent to the segment SR.

From the options provided, the statements that would help to establish triangle PQR is congruent to triangle TSR by the SAS Theorem are:

• QR is congruent to SR (this gives the second pair of congruent sides).
• Angle PRQ is congruent to angle TRS (this provides the angle between the two sides we've established is congruent).

Thus, with QR ≅ SR and Angle PRQ ≅ Angle TRS, we can apply the SAS Congruence Theorem.

In summary, we would need:

• \(QR \cong SR\) and
• Angle PRQ \(\cong\) Angle TRS.

This establishes the congruence of triangles PQR and TSR using the SAS Theorem.

Answer 2

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Answer 3

To state that triangle PQR is congruent to triangle TSR by the SAS (Side-Angle-Side) Congruence Theorem, you would need the following additional information:

1. QR is congruent to SR (this gives a pair of congruent sides),
2. Angle PRQ is congruent to angle TRS (this provides the angle between the two sides we are considering).

So the answer you need is:

• QR is congruent to SR
• Angle PRQ is congruent to angle TRS

With these two pieces of information, you can conclude that triangle PQR is congruent to triangle TSR by the SAS Congruence Theorem.