Given: KL || MN Prove: KM/JM = LN/JL

See the bolded parts below. It corresponds to the blanks we have to fill in for each step.

Step-by-step explanation:

Taking a look at the figure, we see that KL ║ MN, ( which is given, as you can see ) while MJ and NJ act each as a transversal. This is a key point that will help us.

2. For step 2, we see that ∠JKL ≅ ∠JMN, while ∠JLK ≅ ∠JNM. This is true is they are present as corresponding angles, on either transversal. We can fill in this blank with " Corresponding angles. "

3. Therefore, for this 3rd step triangles JKL and JMN will be similar by " angle angle similarity. " After all, ∠JKL ≅ ∠JMN / ∠JLK ≅ ∠JNM.

4. This step can be proved by the " Proportionality of Corresponding Parts in Similar Triangles. " As you can see, the triangles are similar - and hence their parts correspond to one another.

5. I believe you mean step 5 to be " JM = JK + KM and JN = JL + LN. " That being said this is true by the " Partition Postulate, " which states that a whole is composed of it's parts.

6. This step substituted the 5th step into the 4th step. Therefore, it can be stated as " Substitute step 5 ➡ step 4. "

7. And for this last step here you can say " Simplify further. "