geometry unit 3 lesson 9 homework name khloe augustine date 9 - 24 - 25 period 5th complete the two - column proof. given: ∠tkf≅∠kfr prove: ∠vkd≅∠wfg write the statements and reasons in the correct order in the two - column proof. ∠tkd≅∠gfr supplements of congruent angles are congruent. ∠vkd≅∠wfg if two angles form a linear pair, then they are supplementary. tv||wr if two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent. ∠gfr and ∠wfg are supplementary if alternate interior angles are congruent, then the two lines that are intersected by a transversal are parallel.

Question

Accepted Answer

# Explanation:
## Step1: Identify parallel - line condition
Since $\angle TKF\cong\angle KFR$ (alternate interior angles), by the converse of alternate - interior angles theorem, we have $\overline{TV}\parallel\overline{WR}$.
Statement: $\overline{TV}\parallel\overline{WR}$; Reason: If alternate interior angles are congruent, then the two lines that are intersected by a transversal are parallel.
## Step2: Find supplementary angle relationships
$\angle TKD$ and $\angle TKF$ form a linear pair, and $\angle GFR$ and $\angle WFG$ form a linear pair. Also, $\angle TKF\cong\angle KFR$.
$\angle TKD$ and $\angle GFR$ are supplements of congruent angles.
Statement: $\angle TKD\cong\angle GFR$; Reason: Supplements of congruent angles are congruent.
## Step3: Prove the target congruence
$\angle VKD$ and $\angle TKD$ form a linear pair, $\angle WFG$ and $\angle GFR$ form a linear pair.
Since $\angle TKD\cong\angle GFR$, and using the property that if two angles form a linear pair, then they are supplementary, we can prove $\angle VKD\cong\angle WFG$.
Statement: $\angle VKD\cong\angle WFG$; Reason: Supplements of congruent angles are congruent.

# Answer:
| Statement | Reason |
|--|--|
| $\angle TKF\cong\angle KFR$ | Given |
| $\overline{TV}\parallel\overline{WR}$ | If alternate interior angles are congruent, then the two lines that are intersected by a transversal are parallel |
| $\angle TKD\cong\angle GFR$ | Supplements of congruent angles are congruent |
| $\angle VKD\cong\angle WFG$ | Supplements of congruent angles are congruent |

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Identify parallel - line condition

Step2: Find supplementary angle relationships

Step3: Prove the target congruence

Answer:

Statement	Reason
$\overline{TV}\parallel\overline{WR}$	If alternate interior angles are congruent, then the two lines that are intersected by a transversal are parallel
$\angle TKD\cong\angle GFR$	Supplements of congruent angles are congruent
$\angle VKD\cong\angle WFG$	Supplements of congruent angles are congruent