1. KL || MN given
2. angle JKL cog angle JMN and angle JLK cong JNM These are corresponding angles on a traversal of parallel lines.
3. Triangle JKL similar to triangle JMN angle angle similarlity
4. JM/JK = JN/JL Proportionality of corresponding parts of similar triangles
5. JM=JK+KM and JN=JL+LN Collinear segment addition postulate
6. (JK+KM)/JK = (JL+LN)/JL Substitution
7. ... addition of fractinos
8. ... simplify
9. KM/JK = LN/JL Simplify (or Subtraction something)