Question

1 ∠qpr ≅ ∠spt
2 ∠psq ≅ ∠prt
3 overline {ps} ≅ overline {pr}
4 m∠rpt = m∠rps + m∠spt
5 m∠qps = m∠qpr + m∠rps
6 m∠rpt = m∠rps + m∠qpr
7 m∠qps = m∠rpt
8 △prt ≅ △psq
9 overline {qs} ≅ overline {rt}

Explanation:

Step1: Given

No reason needed as it's given in the problem.

Step2: Given

No reason needed as it's given in the problem.

Step3: Given

No reason needed as it's given in the problem.

Step4: Angle - Addition Postulate

The measure of an angle formed by two adjacent angles is the sum of the measures of the two adjacent angles. So, \(m\angle RPT=m\angle RPS + m\angle SPT\).

Step5: Angle - Addition Postulate

The measure of an angle formed by two adjacent angles is the sum of the measures of the two adjacent angles. So, \(m\angle QPS=m\angle QPR + m\angle RPS\).

Step6: Substitution Property of Equality

Since \(\angle QPR\cong\angle SPT\) (from step 1), we can substitute \(m\angle QPR\) for \(m\angle SPT\) in \(m\angle RPT=m\angle RPS + m\angle SPT\) to get \(m\angle RPT=m\angle RPS + m\angle QPR\).

Step7: Substitution Property of Equality

From step 5 (\(m\angle QPS=m\angle QPR + m\angle RPS\)) and step 6 (\(m\angle RPT=m\angle RPS + m\angle QPR\)), by substitution \(m\angle QPS = m\angle RPT\).

Step8: ASA (Angle - Side - Angle) Congruence Postulate

We have \(\angle PSQ\cong\angle PRT\) (step 2), \(\overline{PS}\cong\overline{PR}\) (step 3), and \(m\angle QPS = m\angle RPT\) (step 7). So, \(\triangle PRT\cong\triangle PSQ\) by ASA.

Step9: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle PRT\cong\triangle PSQ\), their corresponding sides \(\overline{QS}\) and \(\overline{RT}\) are congruent.

Answer:

1. Given
2. Given
3. Given
4. Angle - Addition Postulate
5. Angle - Addition Postulate
6. Substitution Property of Equality
7. Substitution Property of Equality
8. ASA (Angle - Side - Angle) Congruence Postulate
9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)