Question

problem 23 given: $overline{km}paralleloverline{jh}$, $overline{km}congoverline{jh}$, $overline{jk}congoverline{kl}$ prove: $overline{hk}paralleloverline{ml}$ 3. $\triangle hjkcong\triangle mkl$ 4. $angle jkhcongangle klm$ 5. $overline{hk}paralleloverline{ml}$ 3. side - angle - side 4. corresponding parts of congruent triangles are congruent (c.p.c.t.c.) 5. click here to insert

Explanation:

Step1: Recall congruence results

Since $\triangle HJK\cong\triangle MKL$ by Side - Angle - Side (SAS) (from given side - length equalities and parallel - line - related angle equalities), we have $\angle JKH\cong\angle KLM$ by Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

Step2: Apply parallel - line theorem

$\angle JKH$ and $\angle KLM$ are alternate - interior angles for lines $\overline{HK}$ and $\overline{ML}$ with transversal $\overline{KL}$. By the Alternate - Interior Angles Converse (if alternate - interior angles are congruent, then the two lines are parallel), we can conclude that $\overline{HK}\parallel\overline{ML}$.

Answer:

1. Alternate - Interior Angles Converse