First, let's draw kite ABCD. In a kite there are two pairs of congruent sides. Let's mark them in our graph.
The diagonals of kite ABCD are BD and AC. We want to show that they are perpendicular. First, point A is equidistant from B and D, and point C is also equidistant from B and D.
AB & = AD
BC &= CD
According to the Converse of the Perpendicular Bisector Theorem, points A and C are on the perpendicular bisector of BD.
Diagonal AC lies on the perpendicular bisector of BD. This tells us that it is perpendicular to diagonal BD.
Paragraph Proof
Let's summarize the proof in one paragraph.
2
&Given:&& ABCD is a kite
& && AB≅AD,CB≅CD
&Prove:&& AC⊥BD
Proof:
Both A and C are equidistant from B and D, so according to the converse of the Perpendicular Bisector Theorem, they are both on the perpendicular bisector of BD. This means that diagonal AC is perpendicular to diagonal BD.
