Question

r is the midpoint of \\(\overline{qs}\\), \\(\overline{su} \cong \overline{rt}\\), and \\(\overline{ru} \cong \overline{qt}\\). complete the proof that \\(\triangle rsu \cong \triangle qrt\\).

	statement	reason
2	\\(\overline{su} \cong \overline{rt}\\)	given
3	\\(\overline{ru} \cong \overline{qt}\\)	given
4	\\(\overline{qr} \cong \overline{rs}\\)
5	\\(\triangle rsu \cong \triangle qrt\\)

Explanation:

Step1: Analyze Statement 4

Since \( R \) is the midpoint of \( \overline{QS} \), by the definition of a midpoint, we know that \( \overline{QR} \cong \overline{RS} \). The reason for this is the "Definition of Midpoint", which states that a midpoint of a segment divides the segment into two congruent segments.

Step2: Analyze Statement 5

To prove \( \triangle RSU \cong \triangle QRT \), we can use the SSS (Side - Side - Side) Congruence Postulate. We have \( \overline{SU} \cong \overline{RT} \) (given), \( \overline{RU} \cong \overline{QT} \) (given), and \( \overline{RS} \cong \overline{QR} \) (from step 1). So the reason for \( \triangle RSU \cong \triangle QRT \) is the "SSS Congruence Postulate", which states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Answer:

• For Statement 4 ( \( \overline{QR} \cong \overline{RS} \) ), the reason is "Definition of Midpoint".
• For Statement 5 ( \( \triangle RSU \cong \triangle QRT \) ), the reason is "SSS Congruence Postulate".