Question

select the correct answer from each drop - down menu. given: kite abdc with diagonals $overline{ad}$ and $overline{bc}$ intersecting at e. prove: $overline{ad}$ bisects $overline{bc}$ determine the missing reasons in the proof.

statement	reason
$overline{cd} \cong \overline{bd}$ and $overline{ac} \cong \overline{ab}$	definition of a kite
$overline{ad} \cong \overline{ad}$	reflexive property of congruence
$\triangle cda \cong \triangle bda$
$\angle cda \cong \angle bda$
$overline{ed} \cong \overline{ed}$	reflexive property of congruence
$\triangle ced \cong \triangle bed$
$overline{ce} \cong \overline{be}$	cpctc
$overline{ad}$ bisects $overline{bc}$	definition of a bisector

(the drop - down options for the reasons of $\triangle cda \cong \triangle bda$ and $\triangle ced \cong \triangle bed$ include sas criterion, hl theorem, asa criterion, sss criterion)

Explanation:

1. For $\triangle CDA \cong \triangle BDA$: We have three pairs of congruent sides: $\overline{CD} \cong \overline{BD}$, $\overline{AC} \cong \overline{AB}$, and $\overline{AD} \cong \overline{AD}$. The congruence rule that uses three pairs of congruent sides is the SSS criterion.
2. For $\triangle CED \cong \triangle BED$: We have $\overline{CD} \cong \overline{BD}$, $\angle CDA \cong \angle BDA$, and $\overline{ED} \cong \overline{ED}$. This is two sides and the included angle, so we use the SAS criterion.

Answer:

• For $\triangle CDA \cong \triangle BDA$: SSS criterion
• For $\triangle CED \cong \triangle BED$: SAS criterion