Ihave no idea what to put statement and reason

Vey interesting !

Look at the drawing:

JX = JY, so the inside triangle is isosceles.

Angle 1  =  Angle 2. 
       Base angles of an isosceles triangle are equal.

Angles (1 + 3)  =  Angles (2 + 4).
       The sum of all the angles on one side of a line is 180 degrees.

Well !  If (1 + 3) = (2 + 4), and  1 = 2, then  Angle 3 = Angle 4 .

Now look at the two skinny, slanty outside triangles:

JX = JY    given
Angle 3 = Angle 4
KX = KY   given

So the two skinny slanty outside triangles are congruent  (side-angle-side)

Then  JK = JL    corresponding parts of congruent triangles.

So the big outside triangle is isosceles, since  JK = JL .

Q.E.D.