Hey Hailey, sorry to hear that you've a problem with this proof. Here I will try to answer.
I wish I could draw a picture here for you to have a better understanding.
Okay, lets start!
ABCD is a parallelogram and BC is the diameter, okay
Now we know a rule of equal alternate interior angle for two parallel lines, right?
Here in our parallelogram side AB is parallel to side CD. therefore, using the above rule we get <ABC=<BCD
again side AC is parallel to side BD. so same rule is applied here.i.e <ACB=<CBD
Now think, as BC is the diameter, we have two triangles. one ABC and other is DBC.
In these two triangle we have two equal angles and one common side, which is BC.
therefore, we can say that two triangles are equal in area.
so we can conclude that AC=BD and AB=CB which implies <A=<C
<ACB=<CBD and <ABC=<BCD.