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: Prove \(\triangle CDA\cong\triangle BDA\)

We have \(CD\cong BD\), \(AC\cong AB\), and \(AD\cong AD\). By the SSS (Side - Side - Side) congruence criterion, \(\triangle CDA\cong\triangle BDA\).

Step2: Prove \(\triangle CED\cong\triangle BED\)

We know \(CD\cong BD\), \(\angle CDA\cong\angle BDA\) (from \(\triangle CDA\cong\triangle BDA\) and CPCTC), and \(ED\cong ED\). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle CED\cong\triangle BED\).

Answer:

For \(\triangle CDA\cong\triangle BDA\), the reason is SSS (Side - Side - Side) congruence criterion. For \(\triangle CED\cong\triangle BED\), the reason is SAS (Side - Angle - Side) congruence criterion.