complete the two - column proof. given: ∠dbv≅∠mpg and ∠tbd≅∠wfb. prove: ∠rfp≅∠mpf. write the statements and reasons in the correct order in the two - column proof. ∠pfr≅∠mpf given ∠tbd≅∠wfb transitive property of congruence ∠pfr≅∠tbd if two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent. ∠tbd≅∠mpf if two parallel lines are intersected by a transversal, then the corresponding angles are congruent. (overleftrightarrow{tv}paralleloverleftrightarrow{mc}) if alternate exterior angles are congruent, then the two lines that are intersected by a transversal are parallel. (overleftrightarrow{tv}paralleloverleftrightarrow{wr}) if corresponding angles are congruent, then the two lines that are intersected by a transversal are parallel.

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# Explanation:
## Step1: Identify given angles
We know $\angle DBV\cong\angle MPG$ and $\angle TBD\cong\angle WFB$ are given.
## Step2: Use corresponding - angle property
Since $\angle TBD\cong\angle WFB$, by the corresponding - angle property of parallel lines (if corresponding angles are congruent, then the two lines that are intersected by a transversal are parallel), we have $\overline{TV}\parallel\overline{WR}$.
## Step3: Use alternate - exterior angle property
Because $\angle DBV\cong\angle MPG$, by the alternate - exterior angle property (if alternate exterior angles are congruent, then the two lines that are intersected by a transversal are parallel), we get $\overline{TV}\parallel\overline{MC}$.
## Step4: Use transitive property of parallel lines
Since $\overline{TV}\parallel\overline{WR}$ and $\overline{TV}\parallel\overline{MC}$, then $\overline{WR}\parallel\overline{MC}$.
## Step5: Use corresponding - angle property again
If $\overline{WR}\parallel\overline{MC}$, then $\angle TBD\cong\angle MPF$ (if two parallel lines are intersected by a transversal, then the corresponding angles are congruent).
## Step6: Use alternate - exterior angle property again
If $\overline{TV}\parallel\overline{WR}$, then $\angle PFR\cong\angle TBD$ (if two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent).
## Step7: Use transitive property of congruence
Since $\angle PFR\cong\angle TBD$ and $\angle TBD\cong\angle MPF$, by the transitive property of congruence, $\angle PFR\cong\angle MPF$.

# Answer:
| Statement | Reason |
|--|--|
| $\angle DBV\cong\angle MPG$ | Given |
| $\angle TBD\cong\angle WFB$ | Given |
| $\overline{TV}\parallel\overline{WR}$ | If corresponding angles are congruent, then the two lines that are intersected by a transversal are parallel |
| $\overline{TV}\parallel\overline{MC}$ | If alternate exterior angles are congruent, then the two lines that are intersected by a transversal are parallel |
| $\overline{WR}\parallel\overline{MC}$ | Transitive property of parallel lines |
| $\angle TBD\cong\angle MPF$ | If two parallel lines are intersected by a transversal, then the corresponding angles are congruent |
| $\angle PFR\cong\angle TBD$ | If two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent |
| $\angle PFR\cong\angle MPF$ | Transitive property of congruence |

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Question

Explanation:

Step1: Identify given angles

Step2: Use corresponding - angle property

Step3: Use alternate - exterior angle property

Step4: Use transitive property of parallel lines

Step5: Use corresponding - angle property again

Step6: Use alternate - exterior angle property again

Step7: Use transitive property of congruence

Answer:

Statement	Reason
$\angle TBD\cong\angle WFB$	Given
$\overline{TV}\parallel\overline{WR}$	If corresponding angles are congruent, then the two lines that are intersected by a transversal are parallel
$\overline{TV}\parallel\overline{MC}$	If alternate exterior angles are congruent, then the two lines that are intersected by a transversal are parallel
$\overline{WR}\parallel\overline{MC}$	Transitive property of parallel lines
$\angle TBD\cong\angle MPF$	If two parallel lines are intersected by a transversal, then the corresponding angles are congruent
$\angle PFR\cong\angle TBD$	If two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent
$\angle PFR\cong\angle MPF$	Transitive property of congruence