Question

an incomplete proof is shown.
given: $overleftrightarrow{st}paralleloverleftrightarrow{rg}$ and $overleftrightarrow{hz}paralleloverleftrightarrow{vw}$
prove: $angle rmh$ is supplementary to $angle nkw$

statement	reason
2. $angle rmhcongangle kpg$	2. blank
3. $angle kpgcongangle wkt$	3. blank
4. $angle rmhcongangle wkt$	4. transitive property of congruence
5. $mangle rmh = mangle wkt$	5. definition of congruence
6. $angle wkt$ and $angle nkw$ are a linear pair	6. definition of linear pair
7. $angle wkt$ and $angle nkw$ are supplementary	7. linear pair postulate
8. $mangle wkt + mangle nkw=180^{circ}$	8. definition of supplementary angles
9. blank + $mangle nkw = 180^{circ}$	9. substitution property of equality
10. $angle rmh$ and $angle nkw$ are supplementary	10. blank

Explanation:

Step1: Identify corresponding - angles

Since $\overleftrightarrow{ST}\parallel\overleftrightarrow{RG}$, $\angle RMH$ and $\angle KPG$ are corresponding angles. So the reason for $\angle RMH\cong\angle KPG$ is "Corresponding - angles postulate".

Step2: Identify vertical - angles

$\angle KPG$ and $\angle WKT$ are vertical angles. So the reason for $\angle KPG\cong\angle WKT$ is "Vertical angles are congruent".

Step3: Substitute

Since $m\angle RMH = m\angle WKT$ (from step 4 and 5 in the proof), in the equation $m\angle WKT+m\angle NKW = 180^{\circ}$, we substitute $m\angle WKT$ with $m\angle RMH$. So the blank in statement 9 is $m\angle RMH$.

Step4: Define supplementary angles

The reason for $\angle RMH$ and $\angle NKW$ are supplementary (statement 10) is "Definition of supplementary angles" (since $m\angle RMH + m\angle NKW=180^{\circ}$).

Answer:

1. Corresponding - angles postulate
2. Vertical angles are congruent
3. $m\angle RMH$
4. Definition of supplementary angles