Question

complete the proof that ∠rqt ≅ ∠qrt.
statement reason
1 ∠pst ≅ ∠spt given
2 qr || ps given
3 ∠spt ≅ ∠qrt alternate interior angles theorem
4 ∠rqt ≅ ∠pst alternate interior angles theorem
5 ∠spt ≅ ∠rqt transitive property of congruence
6 ∠rqt ≅ ∠qrt reflexive property of congruence reflexive property of equality substitution transitive property of congruence transitive property of equality

Explanation:

Step1: Given equal - angled pair

We are given that $\angle PST\cong\angle SPT$.

Step2: Given parallel lines

We are given that $\overleftrightarrow{QR}\parallel\overleftrightarrow{PS}$.

Step3: Use alternate - interior angles

Since $\overleftrightarrow{QR}\parallel\overleftrightarrow{PS}$, by the Alternate Interior Angles Theorem, $\angle SPT\cong\angle QRT$.

Step4: Use alternate - interior angles again

Also, since $\overleftrightarrow{QR}\parallel\overleftrightarrow{PS}$, by the Alternate Interior Angles Theorem, $\angle RQT\cong\angle PST$.

Step5: Apply transitive property

Since $\angle PST\cong\angle SPT$ (Step 1) and $\angle SPT\cong\angle QRT$ (Step 3) and $\angle RQT\cong\angle PST$ (Step 4), by the Transitive Property of Congruence, if $a = b$ and $b = c$ and $d=a$, then $d = c$. So $\angle RQT\cong\angle QRT$.

Answer:

Transitive Property of Congruence