A kite is a quadrilateral with two pairs of adjacent, congruent sides. Prove the two angles between the non-congruent sides are congruent. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted.

let the kite is the quadrilateral ABCD with the two pairs of adjacent congruent sides. (the diagram is also attached) 

Now, its given in question, see from figure,
AB is congruent to AD 
BC is congruent to DC 

Now, let us join the points A & C to form AC ; and points B and D to form BD. 
So, AC is common side  to triangles ABC and ADC. 
So, Because AB ≈AD
BC ≈ DC
And, AC is common, therefore, 
triangle ABC is congruent to triangle ADC 

⇒∠ ABC ≈ ∠ADC (These are the angles between the non-congruent sides)