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Answer :
We start by noting that quadrilateral [tex]\(ABCD\)[/tex] has one pair of opposite sides, [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex], parallel, and the other pair, [tex]\(BC\)[/tex] and [tex]\(DA\)[/tex], also parallel. By drawing diagonal [tex]\(BD\)[/tex], two triangles—[tex]\(\triangle ABD\)[/tex] and [tex]\(\triangle CDB\)[/tex]—are formed.
1. In these triangles, the proof shows that two angles in [tex]\(\triangle ABD\)[/tex] are congruent to two angles in [tex]\(\triangle CDB\)[/tex] because they are alternate interior angles. That is,
[tex]$$ \angle ABD \cong \angle CDB \quad \text{and} \quad \angle BDA \cong \angle DBC. $$[/tex]
2. The diagonal [tex]\(BD\)[/tex] is a common side to both triangles, meaning
[tex]$$ BD \cong BD \quad (\text{reflexive property}). $$[/tex]
3. With two pairs of congruent angles and the common side, the ASA (Angle-Side-Angle) postulate confirms that
[tex]$$ \triangle ABD \cong \triangle CDB. $$[/tex]
4. By the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, corresponding sides in the triangles are congruent. Hence, it follows that:
[tex]$$ AB \cong CD \quad \text{and} \quad BC \cong DA. $$[/tex]
This result shows that the opposite sides of quadrilateral [tex]\(ABCD\)[/tex] are congruent. Therefore, the statement established by the proof is:
[tex]$$\text{"The opposite sides of a parallelogram are congruent."}$$[/tex]
Thus, the correct answer is option 3.
1. In these triangles, the proof shows that two angles in [tex]\(\triangle ABD\)[/tex] are congruent to two angles in [tex]\(\triangle CDB\)[/tex] because they are alternate interior angles. That is,
[tex]$$ \angle ABD \cong \angle CDB \quad \text{and} \quad \angle BDA \cong \angle DBC. $$[/tex]
2. The diagonal [tex]\(BD\)[/tex] is a common side to both triangles, meaning
[tex]$$ BD \cong BD \quad (\text{reflexive property}). $$[/tex]
3. With two pairs of congruent angles and the common side, the ASA (Angle-Side-Angle) postulate confirms that
[tex]$$ \triangle ABD \cong \triangle CDB. $$[/tex]
4. By the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, corresponding sides in the triangles are congruent. Hence, it follows that:
[tex]$$ AB \cong CD \quad \text{and} \quad BC \cong DA. $$[/tex]
This result shows that the opposite sides of quadrilateral [tex]\(ABCD\)[/tex] are congruent. Therefore, the statement established by the proof is:
[tex]$$\text{"The opposite sides of a parallelogram are congruent."}$$[/tex]
Thus, the correct answer is option 3.
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