Question

given: $overline{km}paralleloverline{jh}$, $overline{km}congoverline{jh}$, $overline{jk}congoverline{kl}$
prove: $overline{hk}paralleloverline{ml}$

1. $overline{km}paralleloverline{jh}$, $overline{km}congoverline{jh}$, $overline{jk}congoverline{kl}$
2. given
3. if parallel lines are cut by a transversal, then their corresponding angles are congruent.
4. $\triangle hjkcong\triangle mkl$

4.

Explanation:

Step1: Identify corresponding angles

Since $\overline{KM}\parallel\overline{JH}$ and $\overline{HK}$ is a transversal, $\angle JHK\cong\angle HKM$.

Step2: Prove triangle congruence

We have $\overline{JK}\cong\overline{KL}$, $\angle JKH$ and $\angle LKM$ are vertical angles so $\angle JKH\cong\angle LKM$, and $\overline{KM}\cong\overline{JH}$. By SAS, $\triangle HJK\cong\triangle MKL$.

Step3: Use congruent - triangle properties

Corresponding angles of congruent triangles $\triangle HJK\cong\triangle MKL$ give $\angle HKL\cong\angle MKL$.

Step4: Prove parallel lines

Since $\angle HKL\cong\angle MKL$ (alternate - interior angles), $\overline{HK}\parallel\overline{ML}$.

Answer:

1. $\angle JHK\cong\angle HKM$ (Corresponding - angles are congruent as $\overline{KM}\parallel\overline{JH}$ cut by transversal $\overline{HK}$); 3. By Side - Angle - Side (SAS) congruence criterion ($\overline{JK}\cong\overline{KL}$, $\angle JKH\cong\angle LKM$ (vertical angles), $\overline{KM}\cong\overline{JH}$); 4. $\angle HKL\cong\angle MKL$ (Corresponding angles of congruent triangles $\triangle HJK\cong\triangle MKL$), so $\overline{HK}\parallel\overline{ML}$ (If alternate - interior angles are congruent, then the lines are parallel)