Question

part 3 of 3
complete this proof.
given ( angle b cong angle d ) and ( overline{ac} cong overline{ce} )
prove ( \triangle cab cong \triangle ced )
complete the proof below
statement\treason
( angle b cong angle d ) and ( overline{ac} cong overline{ce} )\tgiven
( angle bca cong angle dce )\tvertical angles theorem
( \triangle cab cong \triangle ced )\tdropdown with options: asa, aas, sas, sss

Explanation:

Step1: Identify known elements

We have $\angle B \cong \angle D$ (given), $\overline{AC} \cong \overline{CE}$ (given), and $\angle BCA \cong \angle DCE$ (vertical angles).

Step2: Match with congruence criteria

The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. Here, we have two angles ($\angle B \cong \angle D$ and $\angle BCA \cong \angle DCE$) and a non - included side ($\overline{AC} \cong \overline{CE}$) for $\triangle CAB$ and $\triangle CED$. So the reason for $\triangle CAB \cong \triangle CED$ is AAS.

Answer:

AAS (The reason for the congruence of $\triangle CAB$ and $\triangle CED$ is the AAS (Angle - Angle - Side) congruence criterion, as we have two congruent angles ($\angle B\cong\angle D$, $\angle BCA\cong\angle DCE$) and a congruent non - included side ($\overline{AC}\cong\overline{CE}$) between the two triangles.)