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)	cpctc
(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

Explanation:

Step1: Analyze $\triangle CDA \cong \triangle BDA$

We have $\overline{CD} \cong \overline{BD}$, $\overline{AC} \cong \overline{AB}$, and $\overline{AD} \cong \overline{AD}$ (reflexive). By SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So $\triangle CDA \cong \triangle BDA$ by SSS.

Step2: Analyze $\triangle CED \cong \triangle BED$

We know $\overline{CD} \cong \overline{BD}$ (from kite definition), $\angle CDA \cong \angle BDA$ (from CPCTC), and $\overline{ED} \cong \overline{ED}$ (reflexive). By SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. So $\triangle CED \cong \triangle BED$ by SAS.

Answer:

For $\triangle CDA \cong \triangle BDA$: SSS (Side - Side - Side) Congruence Criterion
For $\triangle CED \cong \triangle BED$: SAS (Side - Angle - Side) Congruence Criterion