Question

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

Explanation:

Step1: Analyze Statement 5

Since \( U \) is the midpoint of \( \overline{QS} \), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So \( \overline{QU} \cong \overline{SU} \) because \( U \) splits \( \overline{QS} \) into two equal parts. The reason for this is "Definition of Midpoint".

Step2: Analyze Statement 6

\( \angle QUT \) and \( \angle RUS \) are vertical angles. Vertical angles are formed when two lines intersect, and they are congruent. So the reason for \( \angle QUT \cong \angle RUS \) is "Vertical Angles Theorem" (which is already noted, but we confirm the logic here).

Step3: Analyze Statement 7

To prove \( \triangle RSU \cong \triangle TQU \), we can use the ASA (Angle - Side - Angle) congruence criterion. We have:

• \( \angle S \cong \angle Q \) (Given, Statement 4)
• \( \overline{SU} \cong \overline{QU} \) (Statement 5, Definition of Midpoint)
• \( \angle RUS \cong \angle TQU \) (Wait, actually, \( \angle RUS \cong \angle QUT \) (Vertical Angles), and \( \angle S \) and \( \angle Q \) are right angles? Wait, no, from Statements 2 and 3, \( \overline{QT}\perp\overline{QS} \) and \( \overline{QS}\perp\overline{RS} \), so \( \angle Q \) and \( \angle S \) are right angles, so \( \angle Q\cong\angle S \) (All right angles are congruent). Then we have \( \angle S=\angle Q \), \( SU = QU \), and \( \angle RUS=\angle QUT \) (Vertical Angles). So by ASA (Angle - Side - Angle) congruence postulate, \( \triangle RSU\cong\triangle TQU \). The reason for Statement 7 is "ASA (Angle - Side - Angle) Congruence Postulate" (or we can also check the elements: \( \angle S\cong\angle Q \), \( SU\cong QU \), \( \angle RUS\cong\angle QUT \), so ASA applies).

Answer:

• Statement 5 Reason: Definition of Midpoint
• Statement 6 Reason: Vertical Angles Theorem (already correct as per the box, but the logic is vertical angles are congruent)
• Statement 7 Reason: ASA (Angle - Side - Angle) Congruence Postulate

(If we are filling the table:
Statement 5 Reason: Definition of Midpoint
Statement 6 Reason: Vertical Angles Theorem
Statement 7 Reason: ASA (Angle - Side - Angle) Congruence Postulate)