Question

the proof that $\triangle abc \cong \triangle cda$ is shown. given $\overline{ab} \parallel \overline{cd}$ and $\overline{bc} \parallel \overline{da}$ prove: $\triangle abc \cong \triangle cda$ what is the missing reason in the proof? \

$$\begin{tabular}{|c|c|} \\hline statements & reasons \\\\ \\hline 1. $\\overline{ab} \\parallel \\overline{cd}$; $\\overline{bc} \\parallel \\overline{da}$ & 1. given \\\\ \\hline 2. quadrilateral $abcd$ is a $\\parallelogram$ & 2. definition of parallelogram \\\\ \\hline 3. $\\overline{ab} \\cong \\overline{cd}$; $\\overline{bc} \\cong \\overline{da}$ & 3. opposite sides of a parallelogram are $\\cong$ \\\\ \\hline 4. $\\overline{ac} \\cong \\overline{ac}$ & 4. reflexive property \\\\ \\hline 5. $\\triangle abc \\cong \\triangle cda$ & 5.? \\\\ \\hline \\end{tabular}$$

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$$\begin{itemize} \\item perpendicular bisector theorem \\item pythagorean theorem \\item hl theorem \\item sss congruence theorem \\end{itemize}$$

Explanation:

Step1: Identify triangle congruence parts

We have \( AB \cong CD \), \( BC \cong DA \), and \( AC \cong AC \) (reflexive property). These are three pairs of congruent sides.

Step2: Determine congruence theorem

The SSS (Side - Side - Side) congruence theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Here, for \( \triangle ABC \) and \( \triangle CDA \), we have three pairs of congruent sides, so the SSS congruence theorem applies.

Answer:

D. SSS congruence theorem