Question

given: quadrilateral abcd is a kite.
prove: \\( \triangle aed \cong \triangle ced \\)
it is given that quadrilateral abcd is a kite. we know
that \\( \overline{ad} \cong \overline{cd} \\) by the definition of
. by
the kite diagonal theorem, \\( \overline{ac} \\) is
to \\( \overline{bd} \\)
this means that angles aed and ced are right angles.
we also see that \\( \overline{ed} \cong \overline{ed} \\) by the
property. therefore, we have that \\( \triangle aed \cong \triangle ced \\) by
.

Explanation:

• A kite has two distinct pairs of adjacent sides that are equal. So, \(AD\cong CD\) by the definition of a kite.
• The kite diagonal theorem states that one diagonal of a kite is perpendicular to the other diagonal. So, \(AC\) is perpendicular (\(\perp\)) to \(BD\).
• \(ED\cong ED\) by the reflexive property (a segment is congruent to itself).
• For right - angled triangles \(\triangle AED\) and \(\triangle CED\), we have \(AD = CD\) (from the kite definition), \(ED=ED\) (reflexive property), and \(\angle AED=\angle CED = 90^{\circ}\). By the Hypotenuse - Leg (HL) congruence criterion for right - angled triangles, \(\triangle AED\cong\triangle CED\).

Answer:

1. kite
2. perpendicular
3. reflexive
4. HL (Hypotenuse - Leg)