Question

complete the proof that ∠ptq≅∠rts.
statement reason
1 m∠ptq + m∠qtr = 180° angles forming a linear pair sum to 180°
2 m∠qtr + m∠rts = 180°
3 m∠ptq + m∠qtr = m∠qtr + m∠rts transitive property of equality
4 m∠ptq = m∠rts properties of addition, subtraction, multiplication, and division
5 ∠ptq≅∠rts definition of congruence

Explanation:

Step1: Identify linear - pair property

Since $\angle QTR$ and $\angle RTS$ form a straight - line, angles forming a linear pair sum to $180^{\circ}$.

Step2: Use transitive property

Given $m\angle PTQ + m\angle QTR=180^{\circ}$ and $m\angle QTR + m\angle RTS = 180^{\circ}$, by the transitive property of equality, $m\angle PTQ + m\angle QTR=m\angle QTR + m\angle RTS$.

Step3: Apply subtraction property

Subtract $m\angle QTR$ from both sides of the equation $m\angle PTQ + m\angle QTR=m\angle QTR + m\angle RTS$. We get $m\angle PTQ=m\angle RTS$.

Step4: Use congruence definition

Since $m\angle PTQ=m\angle RTS$, by the definition of congruent angles (angles with equal measures are congruent), $\angle PTQ\cong\angle RTS$.

Answer:

The reason for statement 2 is "Angles forming a linear pair sum to $180^{\circ}$" as $\angle QTR$ and $\angle RTS$ form a linear pair.