Question

2 ∠t ≅ ∠r given
3 ∠tsu ≅ ∠qsr vertical angle theorem
4 ∠psu ≅ ∠psq definition of angle bisector
5 m∠pst = m∠psu + m∠tsu substitution
6 m∠psr = m∠psq + m∠qsr vertical angle theorem
7 m∠pst = m∠psq + m∠qsr reflexive property of congruence
8 m∠psr = m∠pst asa
9 overline{ps} ≅ overline{ps} cpctc
10 △pst ≅ △psr
11 overline{rs} ≅ overline{st}

Explanation:

Step1: Analyze the given congruent angles and segments

We have \(\angle T\cong\angle R\), \(\angle TSU\cong\angle QSR\) (Vertical Angle Theorem), \(\angle PSU\cong\angle PSQ\) (Definition of angle bisector), \(m\angle PST = m\angle PSU + m\angle TSU\) (Angle - Addition Postulate), \(m\angle PSR=m\angle PSQ + m\angle QSR\) (Angle - Addition Postulate).
By substitution (since \(m\angle PST = m\angle PSU + m\angle TSU\) and \(m\angle PSR=m\angle PSQ + m\angle QSR\), and \(\angle PSU\cong\angle PSQ\), \(\angle TSU\cong\angle QSR\)), we get \(m\angle PST=m\angle PSQ + m\angle QSR\) and \(m\angle PSR=m\angle PSQ + m\angle QSR\), so \(m\angle PSR = m\angle PST\) (Transitive Property of Equality).
\(\overline{PS}\cong\overline{PS}\) (Reflexive Property of Congruence).

Step2: Use the ASA (Angle - Side - Angle) congruence criterion

For \(\triangle PST\) and \(\triangle PSR\), we have \(\angle T\cong\angle R\), \(\overline{PS}\cong\overline{PS}\), \(m\angle PST=m\angle PSR\). So, \(\triangle PST\cong\triangle PSR\) by ASA (Angle - Side - Angle) congruence criterion.

Step3: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle PST\cong\triangle PSR\), then \(\overline{RS}\cong\overline{ST}\) (CPCTC).

Answer:

1. ASA (Angle - Side - Angle)
2. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)