Question

t is the midpoint of \\( \overline{rs} \\). complete the proof that \\( \angle s \cong \angle r \\).
image of triangle rqs with t on rs, qt perpendicular to rs

statement	reason
2. \\( \overline{qt} \perp \overline{rs} \\)
3. \\( \angle qtr \cong \angle qts \\)
4. \\( \overline{rt} \cong \overline{st} \\)
5. \\( \overline{qt} \cong \overline{qt} \\)
6. \\( \triangle qrt \cong \triangle qst \\)
7. \\( \angle s \cong \angle r \\)

Explanation:

Step1: Midpoint definition

By the definition of a midpoint, if \(T\) is the midpoint of \(\overline{RS}\), then \(RT = ST\). So, \(\overline{RT}\cong\overline{ST}\) (Reason: Definition of midpoint).

Step2: Right angles

Since \(QT\perp RS\), \(\angle QTR\) and \(\angle QTS\) are right angles. Right angles are congruent, so \(\angle QTR\cong\angle QTS\) (Reason: All right angles are congruent).

Step3: Common side

\(\overline{QT}\cong\overline{QT}\) by the reflexive property of congruence (Reason: Reflexive property of congruence).

Step4: SAS congruence

In \(\triangle QRT\) and \(\triangle QST\), we have \(\overline{RT}\cong\overline{ST}\), \(\angle QTR\cong\angle QTS\), and \(\overline{QT}\cong\overline{QT}\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle QRT\cong\triangle QST\) (Reason: SAS congruence).

Step5: Corresponding parts

Since \(\triangle QRT\cong\triangle QST\), their corresponding angles \(\angle S\) and \(\angle R\) are congruent. So, \(\angle S\cong\angle R\) (Reason: Corresponding parts of congruent triangles are congruent).

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Definition of perpendicular (forms right angles)
4. Reason: Definition of midpoint
5. Reason: Reflexive property of congruence
6. Reason: SAS (\(\overline{RT}\cong\overline{ST}\), \(\angle QTR\cong\angle QTS\), \(\overline{QT}\cong\overline{QT}\))
7. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)