Question

u is the midpoint of \\(\overline{rt}\\) and \\(\overline{qs}\\). complete the proof that \\(\triangle qtu \cong \triangle sru\\).

	statement	reason
1	u is the midpoint of \\(\overline{rt}\\)	given
2	u is the midpoint of \\(\overline{qs}\\)	given
3	\\(\overline{ru} \cong \overline{tu}\\)
4	\\(\overline{qu} \cong \overline{su}\\)
5	\\(\angle qut \cong \angle rus\\)
6	\\(\triangle qtu \cong \triangle sru\\)	sas

Explanation:

Step1: Reason for \( \overline{RU} \cong \overline{TU} \)

By the definition of a midpoint, a midpoint divides a segment into two congruent segments. Since \( U \) is the midpoint of \( \overline{RT} \), \( \overline{RU} \) and \( \overline{TU} \) are the two segments formed, so \( \overline{RU} \cong \overline{TU} \) (Definition of Midpoint).

Step2: Reason for \( \overline{QU} \cong \overline{SU} \)

Similarly, as \( U \) is the midpoint of \( \overline{QS} \), by the definition of a midpoint, \( \overline{QU} \) and \( \overline{SU} \) are congruent (Definition of Midpoint).

Step3: Reason for \( \angle QUT \cong \angle RUS \)

\( \angle QUT \) and \( \angle RUS \) are vertical angles. Vertical angles are always congruent, so \( \angle QUT \cong \angle RUS \) (Vertical Angles Theorem).

Answer:

• For statement 3: Reason is "Definition of Midpoint"
• For statement 4: Reason is "Definition of Midpoint"
• For statement 5: Reason is "Vertical Angles Theorem"