Question

given: quadrilateral abcd is a kite. prove: △aed≅△ced. it is given that quadrilateral abcd is a kite. we know that ad≅cd by the definition of. by the kite diagonal theorem, ac is to bd. this means that angles aed and ced are right angles. we also see that ed≅ed by the property. therefore, we have that △aed≅△ced by aas hl sas

Explanation:

Step1: Recall kite - property

In a kite, two pairs of adjacent sides are equal. So, for kite \(ABCD\), \(AD\cong CD\) by the definition of a kite.

Step2: Recall kite - diagonal property

By the kite diagonal theorem, the diagonals of a kite are perpendicular. So, \(AC\) is perpendicular to \(BD\), which means \(\angle AED=\angle CED = 90^{\circ}\).

Step3: Identify common side

We see that \(ED\cong ED\) by the reflexive property.

Step4: Determine congruence criterion

We have a right - angle in both \(\triangle AED\) and \(\triangle CED\), a common side \(ED\), and the hypotenuses \(AD\) and \(CD\) are equal. So, \(\triangle AED\cong\triangle CED\) by the Hypotenuse - Leg (HL) congruence criterion.

Answer:

The blanks should be filled as follows:

1. a kite (already filled conceptually in the reasoning for \(AD\cong CD\))
2. perpendicular
3. reflexive
4. HL