Question

determine the missing reasons in the proof. statement reason abdc is a kite. given \\( \overline { c d } \cong \overline { b d } \\) and \\( \overline { a c } \cong \overline { a b } \\) definition of a kite \\( \overline { a d } \cong \overline { a d } \\) reflexive property of congruence \\( \triangle c d a \cong \triangle b d a \\) \\( \angle c d a \cong \angle b d a \\) cpctc \\( \overline { e d } \cong \overline { e d } \\) reflexive property of congi \\( \triangle c e d \cong \triangle b e d \\) \\( \overline { c e } \cong \overline { b e } \\) sas criterion hl theorem asa criterion sss criterion \\( \overline { a d } \\) bisects \\( \overline { b c } \\)

Explanation:

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

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

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

Since $\triangle CDA\cong\triangle BDA$, then $\angle CDA\cong\angle BDA$ (CPCTC - Corresponding Parts of Congruent Triangles are Congruent). Also, $\overline{ED}\cong\overline{ED}$ (reflexive property). And we know from the kite, $\overline{CD}\cong\overline{BD}$. So, in $\triangle CED$ and $\triangle BED$, $\overline{CD}\cong\overline{BD}$, $\angle CDE\cong\angle BDE$, $\overline{ED}\cong\overline{ED}$. By the SAS (Side - Angle - Side) criterion of congruence (if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), $\triangle CED\cong\triangle BED$.

Answer:

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