The next one will be my last! will someone make sure to explain it in depth so i understand 100% how to do these!

Step-by-step explanation:

Before we start, let's look at what we're trying to prove: that two triangles are congruent.  There are a few ways we can do that: SSS, SAS, ASA, or AAS.  Whichever we choose, we'll need to show that at least one pair of sides is congruent.  We can do that, since we know that H is the midpoint of LM.  So we'll either use ASA or AAS.

1. LG || JM, H is the midpoint of LM

Given

2. LH ≅ HM

Definition of midpoint

3. ∠GLH ≅ ∠JMH

Alternate interior angles theorem

(∠GLH and ∠JMH are alternate interior angles.  Since LG and JM are parallel, the alternate interior angles are congruent.)

4. ∠LHG ≅ ∠MHJ

Vertical angles theorem

(∠LHG and ∠MHJ are vertical angles, which are always congruent.)

5. ΔLGH ≅ ΔMJH

ASA

(We have two pairs of congruent angles, and a pair of congruent sides between them.)

Now, I chose to use ASA.  However, you could use AAS.  Instead of using vertical angles in step 4, we could have used alternate interior angles theorem to show that ∠LGH ≅ ∠MJH.