Question

given $overleftrightarrow{bh}paralleloverleftrightarrow{rp}$, $angle hxwcongangle ywg$ prove $overleftrightarrow{rf}paralleloverleftrightarrow{kg}$
statement
1 $overleftrightarrow{bh}paralleloverleftrightarrow{rf}$
2 $angle hxwcongangle lyf$
3 $angle hxwcongangle ywg$
4 $angle lyfcongangle ywg$
5 $overleftrightarrow{rp}paralleloverleftrightarrow{kg}$
corresponding angles theorem
converse of the corresponding angles theorem
converse of the alternate interior angles theorem
alternate interior angles theorem

Explanation:

Step1: Given parallel lines

Given $\overleftrightarrow{BH}\parallel\overleftrightarrow{RP}$ and $\angle HXW\cong\angle YWG$. Since $\overleftrightarrow{BH}\parallel\overleftrightarrow{RP}$, by the Corresponding - Angles Theorem, if we consider a transversal intersecting $\overleftrightarrow{BH}$ and $\overleftrightarrow{RP}$, we can establish angle - relationships.

Step2: Identify corresponding angles

We know that $\angle HXW$ and $\angle LYF$ are corresponding angles. If we assume a transversal intersecting $\overleftrightarrow{BH}$ and $\overleftrightarrow{RF}$ (where $\overleftrightarrow{BH}\parallel\overleftrightarrow{RF}$ by the given information in the proof - setup). By the Corresponding Angles Theorem, if two parallel lines are cut by a transversal, corresponding angles are congruent.

Step3: Use given angle congruence

We are given that $\angle HXW\cong\angle YWG$.

Step4: Transitive property of congruence

Since $\angle HXW\cong\angle LYF$ and $\angle HXW\cong\angle YWG$, by the transitive property of congruence, $\angle LYF\cong\angle YWG$.

Step5: Apply converse of alternate - interior angles theorem

$\angle LYF$ and $\angle YWG$ are alternate - interior angles. By the Converse of the Alternate Interior Angles Theorem, if alternate - interior angles are congruent, then the two lines $\overleftrightarrow{RF}$ and $\overleftrightarrow{KG}$ are parallel.

Answer:

The reason for step 5 is the Converse of the Alternate Interior Angles Theorem.