Question

u is the midpoint of \\(\overline{rt}\\) and \\(\overline{qs}\\) and \\(\overline{rs} \cong \overline{qt}\\). complete the proof that \\(\triangle rsu \cong \triangle tqu\\).

	statement	reason
2	u is the midpoint of \\(\overline{qs}\\)	given
3	\\(\overline{rs} \cong \overline{qt}\\)	given
4	\\(\overline{ru} \cong \overline{tu}\\)
5	\\(\overline{qu} \cong \overline{su}\\)
6	\\(\triangle rsu \cong \triangle tqu\\)

Explanation:

Step 1: Reason for $\overline{RU} \cong \overline{TU}$

A midpoint divides a segment into two congruent segments. Since \( U \) is the midpoint of \( \overline{RT} \), by the definition of a midpoint, \( \overline{RU} \cong \overline{TU} \).

Step 2: Reason for $\overline{QU} \cong \overline{SU}$

Similarly, \( U \) is the midpoint of \( \overline{QS} \). By the definition of a midpoint, a midpoint divides a segment into two congruent segments, so \( \overline{QU} \cong \overline{SU} \).

Step 3: Reason for $\triangle RSU \cong \triangle TQU$

We have \( \overline{RS} \cong \overline{QT} \) (given), \( \overline{RU} \cong \overline{TU} \) (from step 1), and \( \overline{QU} \cong \overline{SU} \) (from step 2). By the Side - Side - Side (SSS) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So \( \triangle RSU \cong \triangle TQU \) by SSS.

Answer:

1. Reason: Definition of midpoint (A midpoint divides a segment into two congruent segments)
2. Reason: Definition of midpoint (A midpoint divides a segment into two congruent segments)
3. Reason: SSS (Side - Side - Side) Congruence Criterion