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. ?

perpendicular bisector theorem
pythagorean theorem
hl theorem
sss congruence theorem

Explanation:

• 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.
• In this proof, we have \(AB\cong CD\), \(BC\cong DA\), and \(AC\cong AC\) (common side). These are the three pairs of congruent sides for \(\triangle ABC\) and \(\triangle CDA\).
• The perpendicular bisector theorem is about a point on the perpendicular bisector of a segment being equidistant from the segment's endpoints.
• The Pythagorean theorem relates the sides of a right - triangle (\(a^{2}+b^{2}=c^{2}\)).
• The HL (Hypotenuse - Leg) theorem is for right - triangles, which we don't know if \(\triangle ABC\) and \(\triangle CDA\) are right - triangles from the given information.

Answer:

SSS congruence theorem