Final answer:
To prove that AE bisects kite ABCD, the properties of a kite's diagonals, reflexive property of congruence, and CPCTC are used to show the two triangles formed are congruent, meaning AE divides AD and the kite into two congruent parts.
Step-by-step explanation:
To prove that line AE bisects the kite ABCD, we first need to understand that by the definition of a kite, two pairs of adjacent sides are equal. Here, we can assume that AB = AD and CB = CD since AD and CB are the diagonals of the kite intersecting at E. To prove the bisector part, we would need to show that AE = DE, which follows from the fact that the diagonals of a kite are perpendicular and intersect at the bisector, making two right-angle triangles AED and CEB having side ED in common.
The reflexive property of congruence tells us that ED is congruent to itself. By considering the right-angle triangles AED and CEB, and applying CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we can conclude that AE must be equal to DE, thus AE is the bisector. The definition of a bisector is a line that divides a segment into two equal parts, which is exactly what AE does to the diagonal AD in the kite ABCD. Hence, AE not only bisects AD but also bisects the kite into two congruent triangles, demonstrating that AE is the angle bisector.