Pear'l proof was incorrect
As you can see Tonya says this:
∠AKL≅∠GKB
Because of vertical angles theorem. This is true.
Vertical angles are two non adjacent angles formed by intersecting two lines. The intersection forms two pair of vertical angles
On the other hand pearl says
∠AKL≅∠GKB
Because of definition of adjacent angles, this is false. Those angles are not adjacent
