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

ANSWERS:

QR
is congruent to SR

angle PRQ is congruent to angle TRS

PQ is congruent to TS

angle QPR is congruent to angle STR

Answer 1

To state that triangle PQR is congruent to triangle TSR by the SAS (Side-Angle-Side) Congruence Theorem, you need two sides and the included angle of one triangle to be congruent to two sides and the included angle of the other triangle.

Given that R is the midpoint of both QS and PT, this means:

1. \(QR\) is congruent to \(SR\) (because R is the midpoint of segment QS).
2. \(PQ\) is congruent to \(TS\) (because R is the midpoint of segment PT).
3. You also need to establish that the angle included between these sides is congruent, which can be either:
   • \( \angle PRQ \) is congruent to \( \angle TRS \) (angle between sides QR and PQ and sides SR and TS respectively) or
   • \( \angle QPR \) is congruent to \( \angle STR \) (the angle respectively opposite to the congruent sides).

Thus, to use the SAS Congruence Theorem, you would need:

Answer: QR is congruent to SR, PQ is congruent to TS, and either angle PRQ is congruent to angle TRS or angle QPR is congruent to angle STR.

So in a simplified way, you would need the information about one of the angles being congruent, in addition to the congruence of the two pairs of sides.