Question

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

1. $\triangle hjkcong\triangle mkl$
2. click here to insert
3. corresponding parts of congruent tri - angles are congruent (c.p.c.t.c.)
4. $overline{hk}paralleloverline{ml}$
5. click here to insert

Explanation:

Step1: Identify corresponding - angles of parallel lines

Since $\overline{KM}\parallel\overline{JH}$, $\angle HJK$ and $\angle MKL$ are corresponding angles, so $\angle HJK = \angle MKL$.

Step2: Use congruent - triangle property and parallel - line criterion

We know $\triangle HJK\cong\triangle MKL$. By C.P.C.T.C., and since $\angle HJK$ and $\angle MKL$ are corresponding angles for $\overline{HK}$ and $\overline{ML}$, if corresponding angles are equal, the lines are parallel. So $\overline{HK}\parallel\overline{ML}$.

Answer:

1. $\angle HJK=\angle MKL$ (Corresponding - angles of parallel lines are equal, since $\overline{KM}\parallel\overline{JH}$)
2. If $\angle HJK=\angle MKL$ (from step 4) and $\triangle HJK\cong\triangle MKL$ (from step 3), and by C.P.C.T.C., then $\angle HJK$ and $\angle MKL$ are corresponding angles formed by $\overline{HK}$ and $\overline{ML}$ with a transversal. So, $\overline{HK}\parallel\overline{ML}$ (If corresponding - angles are equal, then the lines are parallel)