Question

→qt bisects ∠rqs. complete the proof that △qst ≅ △qrt.
(image of triangle qsr with t on sr, qt drawn, qs and qr marked congruent)
(table with statement and reason columns:
statement 1: qt bisects ∠rqs (reason: given)
statement 2: qs ≅ qr (reason: given)
statement 3: ∠rqt ≅ ∠sqt
statement 4: qt ≅ qt
statement 5: △qst ≅ △qrt
reasons for 3,4,5 are blank)

Explanation:

Step1: Definition of angle bisector

Since \( \overrightarrow{QT} \) bisects \( \angle RQS \), by the definition of an angle bisector, \( \angle RQT=\angle SQT \).

Step2: Reflexive property of congruence

For any segment \( \overline{QT} \), \( \overline{QT}\cong\overline{QT} \) (reflexive property: a segment is congruent to itself).

Step3: SAS (Side - Angle - Side) congruence criterion

We are given \( \overline{QS}\cong\overline{QR} \) (from the problem statement), \( \angle RQT=\angle SQT \) (from Step 1), and \( \overline{QT}\cong\overline{QT} \) (from Step 2).
By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
In \( \triangle QST \) and \( \triangle QRT \), \( \overline{QS}\cong\overline{QR} \), \( \angle SQT\cong\angle RQT \), \( \overline{QT}\cong\overline{QT} \), so \( \triangle QST\cong\triangle QRT \).

Answer:

1. \( \angle RQT=\angle SQT \) (Reason: Definition of angle bisector)
2. \( \overline{QT}\cong\overline{QT} \) (Reason: Reflexive property of congruence)
3. \( \triangle QST\cong\triangle QRT \) (Reason: SAS (Side - Angle - Side) congruence criterion)