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?

statements	reasons
2. quadrilateral \\( abcd \\) is a \\( \square \\)	2. definition of parallelogram
3. \\( \overline{ab} \cong \overline{cd} \\); \\( \overline{bc} \cong \overline{da} \\)	3. opposite sides of a parallelogram are \\( \cong \\)
4. \\( \overline{ac} \cong \overline{ac} \\)	4. reflexive property
5. \\( \triangle abc \cong \triangle cda \\)	5. ?

\\( \bigcirc \\) perpendicular bisector theorem
\\( \bigcirc \\) pythagorean theorem
\\( \bigcirc \\) hl theorem
\\( \bigcirc \\) sss congruence theorem

Explanation:

Step1: Recall the SSS 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 two triangles are congruent.

Step2: Check the congruent sides

From the proof:

• We know that \(\overline{AB}\cong\overline{CD}\) (from statement 3)
• \(\overline{BC}\cong\overline{DA}\) (from statement 3)
• \(\overline{AC}\cong\overline{AC}\) (from statement 4, reflexive property)

So, in \(\triangle ABC\) and \(\triangle CDA\), we have three pairs of congruent sides.

Answer:

SSS congruence theorem