High School

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Which statement is sufficient evidence that \( \triangle DEF \) is congruent to \( \triangle ABC \)?

1. \( AB = DE \) and \( BC = EF \)

2. \( \angle D \cong \angle A \), \( \angle B \cong \angle E \), \( \angle C \cong \angle F \)

3. There is a sequence of rigid motions that maps \( \overline{AB} \) onto \( \overline{DE} \), \( \overline{BC} \) onto \( \overline{EF} \), and \( \overline{AC} \) onto \( \overline{DF} \).

4. There is a sequence of rigid motions that maps point A onto point D, \( \overline{AB} \) onto \( \overline{DE} \), and \( \angle B \) onto \( \angle E \).

Answer :

To determine which statement is sufficient evidence to prove that [tex]\triangle DEF[/tex] is congruent to [tex]\triangle ABC[/tex], we need to consider the criteria for triangle congruence. Triangles are congruent if all corresponding sides and angles are equal.

  1. Statement 1: [tex]AB = DE[/tex] and [tex]BC = EF[/tex]

    • This information provides two pairs of equal sides, but to establish triangle congruence, we would also need the third side or another pair of angles to match. Therefore, this is not sufficient to prove congruence by itself.

  2. Statement 2: [tex]\angle D \cong \angle A[/tex], [tex]\angle B \cong \angle E[/tex], [tex]\angle C \cong \angle F[/tex]

    • This statement provides information about the angles, but without any information about the side lengths, we cannot use it to confirm triangle congruence. Therefore, this is not sufficient for congruence.

  3. Statement 3: There is a sequence of rigid motions that maps [tex]\overline{AB}[/tex] onto [tex]\overline{DE}[/tex], [tex]\overline{BC}[/tex] onto [tex]\overline{EF}[/tex], and [tex]\overline{AC}[/tex] onto [tex]\overline{DF}[/tex].

    • Rigid motions (translations, rotations, or reflections) preserve distances and angles. If a sequence of rigid motions maps all corresponding sides to each other, then the triangles are congruent. Hence, this statement is sufficient evidence of congruence.

  4. Statement 4: There is a sequence of rigid motions that maps point A onto point D, [tex]\overline{AB}[/tex] onto [tex]\overline{DE}[/tex], and [tex]\angle B[/tex] onto [tex]\angle E[/tex].

    • This statement covers a point and a side mapping, but only describes mapping one angle, not all sides or the entire triangle. Therefore, it is insufficient for proving congruence.

The correct choice is: Statement 3, since it provides a complete mapping of all corresponding sides, ensuring the triangles are congruent through rigid motions.

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