Question

what is the missing reason in the two - column proof?
given: \\( \overrightarrow { a c } \\) bisects \\( \angle d a b \\) and \\( \overrightarrow { c a } \\) bisects \\( \angle d c b \\)
prove: \\( \triangle d a c \cong \triangle b a c \\)

\\( \

$$\begin{array} { | l | l | } \\hline \\text { statements } & \\text { reasons } \\\\ \\hline 1. \\overrightarrow { a c } \\text { bisects } \\angle d a b & 1. \\text { given } \\\\ \\hline 2. \\angle d a c \\cong \\angle b a c & 2. \\text { definition of angle bisector } \\\\ \\hline 3. \\overline { a c } \\cong \\overline { a c } & 3. \\text { reflexive property } \\\\ \\hline 4. \\overrightarrow { c a } \\text { bisects } \\angle d c b & 4. \\text { given } \\\\ \\hline 5. \\angle d c a \\cong \\angle b c a & 5. \\text { definition of angle bisector } \\\\ \\hline 6. \\triangle d a c \\cong \\triangle b a c & 6. \\text {? } \\\\ \\hline \\end{array}$$

\\)
asa postulate
sss postulate
aas theorem
sas postulate

Explanation:

Step1: Recall triangle congruence postulates

We have two angles ($\angle DAC\cong\angle BAC$, $\angle DCA\cong\angle BCA$) and the included side ($\overline{AC}\cong\overline{AC}$) between them.

Step2: Match with postulate

The ASA (Angle - Side - Angle) Postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Here, in $\triangle DAC$ and $\triangle BAC$, $\angle DAC\cong\angle BAC$, $\overline{AC}\cong\overline{AC}$, $\angle DCA\cong\angle BCA$ satisfy the ASA Postulate.

Answer:

ASA Postulate