Which postulate or theorem proves that △abc and △cda are congruent?

ASA congruence postulate

Step-by-step explanation:

The ASA (Angle-Side-Angle) congruence postulate states that if two triangles have two angles and the included side (the side between the vertices of the two angles) congruent, then the triangles are congruent.

The two congruent angles are ∠ACB  ≅ ∠CAD and ∠CAB ≅ ∠DCA. The included side is segment AC

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)