Question

statement reason

1. (overleftrightarrow{fh}paralleloverleftrightarrow{tv}) given
2. (overleftrightarrow{wy}paralleloverleftrightarrow{fh}) given
3. (angle sxycongangle hgs)
4. (mangle hgs + mangle vuz=180^{circ})
5. (mangle vuz + mangle sxy = 180^{circ}) substitution

Explanation:

Step1: Identify corresponding - angles

Since $\overleftrightarrow{FH}\parallel\overleftrightarrow{TV}$ and $\overleftrightarrow{WY}\parallel\overleftrightarrow{FH}$, by the transitive property of parallel lines, $\overleftrightarrow{WY}\parallel\overleftrightarrow{TV}$. $\angle SXY$ and $\angle HGS$ are corresponding - angles. When two parallel lines are cut by a transversal, corresponding angles are congruent. So the reason for $\angle SXY\cong\angle HGS$ is "Corresponding angles postulate".

Step2: Identify same - side interior angles

$\angle HGS$ and $\angle VUZ$ are same - side interior angles. When two parallel lines ($\overleftrightarrow{FH}$ and $\overleftrightarrow{TV}$) are cut by a transversal, same - side interior angles are supplementary. So the reason for $m\angle HGS + m\angle VUZ=180^{\circ}$ is "Same - side interior angles postulate".

Answer:

1. Corresponding angles postulate
2. Same - side interior angles postulate