Question

suppose \\( \overline { b d } \cong \overline { c a } \\) and \\( \overline { a b } \cong \overline { c d } \\). can you use the sss postulate or the sas postulate to prove \\( \triangle a b d \cong \triangle d c a \\)?
both apply
by sss only
neither apply
by sas only

Explanation:

Step1: Analyze SSS Postulate

SSS (Side - Side - Side) Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
We have \(\overline{AB}\cong\overline{CD}\), \(\overline{BD}\cong\overline{CA}\), and \(\overline{AD}\cong\overline{DA}\) (common side). So, \(AB = CD\), \(BD=CA\), \(AD = DA\). By SSS, \(\triangle ABD\cong\triangle DCA\).

Step2: Analyze SAS Postulate

SAS (Side - Angle - Side) Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
We have \(\overline{AB}\cong\overline{CD}\), \(\overline{AD}\cong\overline{DA}\), but there is no information about the included angles (\(\angle BAD\) and \(\angle CDA\)) being congruent. So, we cannot use SAS.

Answer:

by SSS only