Question

select all the answers needed.
what conditions would prove quadrilateral abcd is a kite?
ab ⊥ bd and ac ⊥ cd
ab = cd and ac = bd
ab || cd and ac || bd
ab = bc and ad = cd
ac ⊥ bd

Explanation:

Step1: Recall the definition of a kite

A kite is a quadrilateral with two distinct pairs of adjacent sides equal.

Step2: Analyze each option

• Option 1: $\overline{AB}\perp\overline{BD}$ and $\overline{AC}\perp\overline{CD}$ only gives information about perpendicularity, not about the sides being equal. So this does not prove it's a kite.
• Option 2: $AB = CD$ and $AC = BD$ does not satisfy the adjacent - side - equal property of a kite.
• Option 3: $\overline{AB}\parallel\overline{CD}$ and $\overline{AC}\parallel\overline{BD}$ describes a parallelogram (if both pairs of opposite sides are parallel) or other parallelogram - like figures, not a kite.
• Option 4: $AB = BC$ and $AD = CD$ satisfies the two distinct pairs of adjacent sides equal property of a kite.
• Option 5: $\overline{AC}\perp\overline{BD}$ is a property of a kite (the diagonals are perpendicular), but it is not sufficient on its own to prove a quadrilateral is a kite. For example, a rhombus (a special type of kite) has perpendicular diagonals, but there are other quadrilaterals with perpendicular diagonals that are not kites.

Answer:

$AB = BC$ and $AD = CD$