Question

problem 23
given: $overline{km}paralleloverline{jh}$, $overline{km}congoverline{jh}$, $overline{jk}congoverline{kl}$
prove: $overline{hk}paralleloverline{ml}$

1. $\triangle hjkcong\triangle mkl$
2. $angle jkhcongangle klm$
3. $overline{hk}paralleloverline{ml}$

Explanation:

Step1: Identify congruent triangles

We have $\overline{KM}\parallel\overline{JH}$, so $\angle JHK=\angle HKM$. With $\overline{KM}\cong\overline{JH}$ and $\overline{JK}\cong\overline{KL}$, by SAS, $\triangle HJK\cong\triangle MKL$.

Step2: Use CPCTC

From $\triangle HJK\cong\triangle MKL$, we get $\angle JKH\cong\angle KLM$ by CPCTC.

Step3: Prove parallel lines

$\angle JKH$ and $\angle KLM$ are alternate - interior angles for $\overline{HK}$ and $\overline{ML}$ with transversal $\overline{KL}$. Since they are congruent, $\overline{HK}\parallel\overline{ML}$.

Answer:

1. Reason: Since $\overline{KM}\parallel\overline{JH}$, then $\angle JHK=\angle HKM$ (alternate - interior angles are congruent). Also, given $\overline{KM}\cong\overline{JH}$, $\overline{JK}\cong\overline{KL}$, by the Side - Angle - Side (SAS) congruence criterion, $\triangle HJK\cong\triangle MKL$.
2. Reason: Since $\angle JKH\cong\angle KLM$ (by CPCTC from $\triangle HJK\cong\triangle MKL$), and these are alternate - interior angles for lines $\overline{HK}$ and $\overline{ML}$ with transversal $\overline{KL}$, then $\overline{HK}\parallel\overline{ML}$ (if alternate - interior angles are congruent, then the lines are parallel).