Question

given: (overline{qs}) bisects (angle tqr), (overline{tq} cong overline{rq}). prove: (\triangle qrs cong \triangle qts)
complete the missing parts of the paragraph proof.
proof:
we know that segment (qs) bisects angle (tqr) because (\boxed{quad}). by the definition of angle bisector, angle (tqs) is congruent to angle (\boxed{quad}). we see that segment (qs) is congruent to segment (sq) by (\boxed{quad}). therefore, we can conclude that triangles (qrs) and (qts) are congruent by (\boxed{quad}).

Explanation:

Step1: Analyze Angle Bisector

Since \(\overline{QS}\) bisects \(\angle TQR\), by the definition of an angle bisector, \(\angle TQS\cong\angle RQS\).

Step2: Identify Common Side

Segment \(QS\) is congruent to itself (reflexive property), so \(\overline{QS}\cong\overline{SQ}\).

Step3: Use Given Congruent Sides

We know \(\overline{TQ}\cong\overline{RQ}\) (given).

Step4: Apply SAS Congruence

Now we have two sides and the included angle congruent: \(\overline{TQ}\cong\overline{RQ}\), \(\angle TQS\cong\angle RQS\), \(\overline{QS}\cong\overline{SQ}\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle QRS\cong\triangle QTS\).

Answer:

1. (First blank) the problem states it (given)
2. (Second blank) \(\angle RQS\)
3. (Third blank) reflexive property (or "itself")
4. (Fourth blank) SAS (Side - Angle - Side) congruence criterion